Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Percentage Accurate: 68.1% → 88.5%

Time: 13.4s

Alternatives: 21

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) (- t x)) (- a z))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - z) * (t - x)) / (a - z))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

Python

def code(x, y, z, t, a):
	return x + (((y - z) * (t - x)) / (a - z))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - z) * (t - x)) / (a - z));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) (- t x)) (- a z))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - z) * (t - x)) / (a - z))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

Python

def code(x, y, z, t, a):
	return x + (((y - z) * (t - x)) / (a - z))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - z) * (t - x)) / (a - z));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+138}:\\ \;\;\;\;t + \frac{y - a}{\frac{z}{x - t}}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(y - a\right) \cdot \frac{x - t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.8e+138)
   (+ t (/ (- y a) (/ z (- x t))))
   (if (<= z 6.2e+168)
     (fma (- t x) (/ (- y z) (- a z)) x)
     (+ t (* (- y a) (/ (- x t) z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.8e+138) {
		tmp = t + ((y - a) / (z / (x - t)));
	} else if (z <= 6.2e+168) {
		tmp = fma((t - x), ((y - z) / (a - z)), x);
	} else {
		tmp = t + ((y - a) * ((x - t) / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.8e+138)
		tmp = Float64(t + Float64(Float64(y - a) / Float64(z / Float64(x - t))));
	elseif (z <= 6.2e+168)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / Float64(a - z)), x);
	else
		tmp = Float64(t + Float64(Float64(y - a) * Float64(Float64(x - t) / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.8e+138], N[(t + N[(N[(y - a), $MachinePrecision] / N[(z / N[(x - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.2e+168], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(t + N[(N[(y - a), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.80000000000000022e138

   1. Initial program 10.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 57.8

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 57.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      7. div-sub N/A

         \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. associate-/l* N/A

         \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      9. associate-/l* N/A

         \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      14. --lowering--.f64 88.2

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Simplified 88.2%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto t - \color{blue}{\left(y - a\right) \cdot \frac{t - x}{z}} \]

      2. clear-num N/A

         \[\leadsto t - \left(y - a\right) \cdot \color{blue}{\frac{1}{\frac{z}{t - x}}} \]

      3. un-div-inv N/A

         \[\leadsto t - \color{blue}{\frac{y - a}{\frac{z}{t - x}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto t - \color{blue}{\frac{y - a}{\frac{z}{t - x}}} \]

      5. --lowering--.f64 N/A

         \[\leadsto t - \frac{\color{blue}{y - a}}{\frac{z}{t - x}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto t - \frac{y - a}{\color{blue}{\frac{z}{t - x}}} \]

      7. --lowering--.f64 88.3

         \[\leadsto t - \frac{y - a}{\frac{z}{\color{blue}{t - x}}} \]

   9. Applied egg-rr 88.3%

      \[\leadsto t - \color{blue}{\frac{y - a}{\frac{z}{t - x}}} \]

   if -6.80000000000000022e138 < z < 6.19999999999999993e168

   1. Initial program 82.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 92.1

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 92.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   if 6.19999999999999993e168 < z

   1. Initial program 29.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 47.1

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 47.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      7. div-sub N/A

         \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. associate-/l* N/A

         \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      9. associate-/l* N/A

         \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      14. --lowering--.f64 86.9

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Simplified 86.9%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+138}:\\ \;\;\;\;t + \frac{y - a}{\frac{z}{x - t}}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(y - a\right) \cdot \frac{x - t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 43.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+80}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -2.65 \cdot 10^{-128}:\\ \;\;\;\;\frac{\left(y - a\right) \cdot x}{z}\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-221}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-26}:\\ \;\;\;\;x - \frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e+80)
   t
   (if (<= z -2.65e-128)
     (/ (* (- y a) x) z)
     (if (<= z -1.8e-221)
       (* t (/ y (- a z)))
       (if (<= z 7.4e-26) (- x (/ (* y x) a)) (* (/ t z) (- z y)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+80) {
		tmp = t;
	} else if (z <= -2.65e-128) {
		tmp = ((y - a) * x) / z;
	} else if (z <= -1.8e-221) {
		tmp = t * (y / (a - z));
	} else if (z <= 7.4e-26) {
		tmp = x - ((y * x) / a);
	} else {
		tmp = (t / z) * (z - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.3d+80)) then
        tmp = t
    else if (z <= (-2.65d-128)) then
        tmp = ((y - a) * x) / z
    else if (z <= (-1.8d-221)) then
        tmp = t * (y / (a - z))
    else if (z <= 7.4d-26) then
        tmp = x - ((y * x) / a)
    else
        tmp = (t / z) * (z - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+80) {
		tmp = t;
	} else if (z <= -2.65e-128) {
		tmp = ((y - a) * x) / z;
	} else if (z <= -1.8e-221) {
		tmp = t * (y / (a - z));
	} else if (z <= 7.4e-26) {
		tmp = x - ((y * x) / a);
	} else {
		tmp = (t / z) * (z - y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e+80:
		tmp = t
	elif z <= -2.65e-128:
		tmp = ((y - a) * x) / z
	elif z <= -1.8e-221:
		tmp = t * (y / (a - z))
	elif z <= 7.4e-26:
		tmp = x - ((y * x) / a)
	else:
		tmp = (t / z) * (z - y)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e+80)
		tmp = t;
	elseif (z <= -2.65e-128)
		tmp = Float64(Float64(Float64(y - a) * x) / z);
	elseif (z <= -1.8e-221)
		tmp = Float64(t * Float64(y / Float64(a - z)));
	elseif (z <= 7.4e-26)
		tmp = Float64(x - Float64(Float64(y * x) / a));
	else
		tmp = Float64(Float64(t / z) * Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.3e+80)
		tmp = t;
	elseif (z <= -2.65e-128)
		tmp = ((y - a) * x) / z;
	elseif (z <= -1.8e-221)
		tmp = t * (y / (a - z));
	elseif (z <= 7.4e-26)
		tmp = x - ((y * x) / a);
	else
		tmp = (t / z) * (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e+80], t, If[LessEqual[z, -2.65e-128], N[(N[(N[(y - a), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, -1.8e-221], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.4e-26], N[(x - N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(t / z), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.30000000000000004e80

   1. Initial program 28.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 52.7%

      \[\leadsto \color{blue}{t} \]

   if -2.30000000000000004e80 < z < -2.6499999999999999e-128

   1. Initial program 74.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 82.6

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      7. div-sub N/A

         \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. associate-/l* N/A

         \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      9. associate-/l* N/A

         \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      14. --lowering--.f64 53.4

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Simplified 53.4%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot \left(y - a\right)}{z}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(y - a\right)}{z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(y - a\right)}}{z} \]

      3. --lowering--.f64 34.2

         \[\leadsto \frac{x \cdot \color{blue}{\left(y - a\right)}}{z} \]

   10. Simplified 34.2%

       \[\leadsto \color{blue}{\frac{x \cdot \left(y - a\right)}{z}} \]

   if -2.6499999999999999e-128 < z < -1.80000000000000006e-221

   1. Initial program 83.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      4. --lowering--.f64 56.7

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   5. Simplified 56.7%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{t \cdot y}{a - z}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{y}{a - z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \frac{y}{a - z}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto t \cdot \color{blue}{\frac{y}{a - z}} \]

      4. --lowering--.f64 72.6

         \[\leadsto t \cdot \frac{y}{\color{blue}{a - z}} \]

   8. Simplified 72.6%

      \[\leadsto \color{blue}{t \cdot \frac{y}{a - z}} \]

   if -1.80000000000000006e-221 < z < 7.3999999999999997e-26

   1. Initial program 94.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{y - z}{a - z}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y - z}{a - z}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(y - z\right)}{a - z}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - z\right) \cdot x}}{a - z}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \frac{x}{a - z}}\right)\right) + 1 \cdot x \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{x}{a - z}} + 1 \cdot x \]

      10. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(y - z\right)\right)} \cdot \frac{x}{a - z} + 1 \cdot x \]

      11. *-lft-identity N/A

          \[\leadsto \left(-1 \cdot \left(y - z\right)\right) \cdot \frac{x}{a - z} + \color{blue}{x} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - z\right), \frac{x}{a - z}, x\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - z\right)\right)}, \frac{x}{a - z}, x\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right), \frac{x}{a - z}, x\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + y\right)}\right), \frac{x}{a - z}, x\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, \frac{x}{a - z}, x\right) \]

      17. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y}, \frac{x}{a - z}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z} - y, \frac{x}{a - z}, x\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - y}, \frac{x}{a - z}, x\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - y, \color{blue}{\frac{x}{a - z}}, x\right) \]

      21. --lowering--.f64 66.5

          \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{a - z}}, x\right) \]

   5. Simplified 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - y, \frac{x}{a - z}, x\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x \cdot y}{a}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{a}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{x \cdot y}{a}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{x \cdot y}{a}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{x \cdot y}{a}} \]

      5. *-lowering-*.f64 60.7

         \[\leadsto x - \frac{\color{blue}{x \cdot y}}{a} \]

   8. Simplified 60.7%

      \[\leadsto \color{blue}{x - \frac{x \cdot y}{a}} \]

   if 7.3999999999999997e-26 < z

   1. Initial program 51.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      4. --lowering--.f64 45.9

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   5. Simplified 45.9%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{-1 \cdot z}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 38.4

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{-z}} \]

   8. Simplified 38.4%

      \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{-z}} \]
   9. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{-1 \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{z \cdot -1}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{t}{z} \cdot \frac{y - z}{-1}} \]

      4. div-inv N/A

         \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{-1}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{t}{z} \cdot \left(\left(y - z\right) \cdot \color{blue}{-1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{t}{z} \cdot \left(\left(y - z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right) \cdot 1\right)\right)} \]

      8. *-rgt-identity N/A

         \[\leadsto \frac{t}{z} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \frac{t}{z} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)} \]

      11. remove-double-neg N/A

          \[\leadsto \frac{t}{z} \cdot \left(\left(\mathsf{neg}\left(y\right)\right) + \color{blue}{z}\right) \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)} \]

      13. sub-neg N/A

          \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(z - y\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{t}{z} \cdot \left(z - y\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{t}{z}} \cdot \left(z - y\right) \]

      16. --lowering--.f64 44.7

          \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(z - y\right)} \]

   10. Applied egg-rr 44.7%

       \[\leadsto \color{blue}{\frac{t}{z} \cdot \left(z - y\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+80}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -2.65 \cdot 10^{-128}:\\ \;\;\;\;\frac{\left(y - a\right) \cdot x}{z}\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-221}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-26}:\\ \;\;\;\;x - \frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+51}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, 0\right)}{z}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-234}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.98 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.8e+51)
   t
   (if (<= z -5.5e-127)
     (/ (fma y x 0.0) z)
     (if (<= z 1.65e-234)
       (* t (/ y a))
       (if (<= z 1.98e-19) (fma z (/ x a) x) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.8e+51) {
		tmp = t;
	} else if (z <= -5.5e-127) {
		tmp = fma(y, x, 0.0) / z;
	} else if (z <= 1.65e-234) {
		tmp = t * (y / a);
	} else if (z <= 1.98e-19) {
		tmp = fma(z, (x / a), x);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.8e+51)
		tmp = t;
	elseif (z <= -5.5e-127)
		tmp = Float64(fma(y, x, 0.0) / z);
	elseif (z <= 1.65e-234)
		tmp = Float64(t * Float64(y / a));
	elseif (z <= 1.98e-19)
		tmp = fma(z, Float64(x / a), x);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.8e+51], t, If[LessEqual[z, -5.5e-127], N[(N[(y * x + 0.0), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.65e-234], N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.98e-19], N[(z * N[(x / a), $MachinePrecision] + x), $MachinePrecision], t]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.7999999999999997e51 or 1.98000000000000001e-19 < z

   1. Initial program 42.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 45.0%

      \[\leadsto \color{blue}{t} \]

   if -3.7999999999999997e51 < z < -5.50000000000000036e-127

   1. Initial program 80.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{y - z}{a - z}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y - z}{a - z}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(y - z\right)}{a - z}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - z\right) \cdot x}}{a - z}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \frac{x}{a - z}}\right)\right) + 1 \cdot x \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{x}{a - z}} + 1 \cdot x \]

      10. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(y - z\right)\right)} \cdot \frac{x}{a - z} + 1 \cdot x \]

      11. *-lft-identity N/A

          \[\leadsto \left(-1 \cdot \left(y - z\right)\right) \cdot \frac{x}{a - z} + \color{blue}{x} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - z\right), \frac{x}{a - z}, x\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - z\right)\right)}, \frac{x}{a - z}, x\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right), \frac{x}{a - z}, x\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + y\right)}\right), \frac{x}{a - z}, x\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, \frac{x}{a - z}, x\right) \]

      17. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y}, \frac{x}{a - z}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z} - y, \frac{x}{a - z}, x\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - y}, \frac{x}{a - z}, x\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - y, \color{blue}{\frac{x}{a - z}}, x\right) \]

      21. --lowering--.f64 52.3

          \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{a - z}}, x\right) \]

   5. Simplified 52.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - y, \frac{x}{a - z}, x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{-1 \cdot z}}, x\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{\mathsf{neg}\left(z\right)}}, x\right) \]

      2. neg-lowering-neg.f64 28.8

         \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{-z}}, x\right) \]

   8. Simplified 28.8%

      \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{-z}}, x\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(x + -1 \cdot x\right)}{z}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(x + -1 \cdot x\right)}{z}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot x} + z \cdot \left(x + -1 \cdot x\right)}{z} \]

       3. distribute-rgt1-in N/A

          \[\leadsto \frac{y \cdot x + z \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot x\right)}}{z} \]

       4. metadata-eval N/A

          \[\leadsto \frac{y \cdot x + z \cdot \left(\color{blue}{0} \cdot x\right)}{z} \]

       5. mul0-lft N/A

          \[\leadsto \frac{y \cdot x + z \cdot \color{blue}{0}}{z} \]

       6. mul0-rgt N/A

          \[\leadsto \frac{y \cdot x + \color{blue}{0}}{z} \]

       7. accelerator-lowering-fma.f64 32.7

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, 0\right)}}{z} \]

   11. Simplified 32.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, x, 0\right)}{z}} \]

   if -5.50000000000000036e-127 < z < 1.65000000000000007e-234

   1. Initial program 88.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      4. --lowering--.f64 45.8

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   5. Simplified 45.8%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot t}}{a} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot t}}{a} \]

      4. --lowering--.f64 37.7

         \[\leadsto \frac{\color{blue}{\left(y - z\right)} \cdot t}{a} \]

   8. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot t}{a}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{y - z}{a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \frac{y - z}{a}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a}} \]

      5. --lowering--.f64 49.1

         \[\leadsto t \cdot \frac{\color{blue}{y - z}}{a} \]

   10. Applied egg-rr 49.1%

       \[\leadsto \color{blue}{t \cdot \frac{y - z}{a}} \]

   11. Taylor expanded in y around inf

       \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 47.0

          \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]

   13. Simplified 47.0%

       \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]

   if 1.65000000000000007e-234 < z < 1.98000000000000001e-19

   1. Initial program 97.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{y - z}{a - z}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y - z}{a - z}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(y - z\right)}{a - z}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - z\right) \cdot x}}{a - z}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \frac{x}{a - z}}\right)\right) + 1 \cdot x \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{x}{a - z}} + 1 \cdot x \]

      10. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(y - z\right)\right)} \cdot \frac{x}{a - z} + 1 \cdot x \]

      11. *-lft-identity N/A

          \[\leadsto \left(-1 \cdot \left(y - z\right)\right) \cdot \frac{x}{a - z} + \color{blue}{x} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - z\right), \frac{x}{a - z}, x\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - z\right)\right)}, \frac{x}{a - z}, x\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right), \frac{x}{a - z}, x\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + y\right)}\right), \frac{x}{a - z}, x\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, \frac{x}{a - z}, x\right) \]

      17. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y}, \frac{x}{a - z}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z} - y, \frac{x}{a - z}, x\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - y}, \frac{x}{a - z}, x\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - y, \color{blue}{\frac{x}{a - z}}, x\right) \]

      21. --lowering--.f64 71.9

          \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{a - z}}, x\right) \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - y, \frac{x}{a - z}, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{z}, \frac{x}{a - z}, x\right) \]
   7. Step-by-step derivation

   8. Simplified 52.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z}, \frac{x}{a - z}, x\right) \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x}{a}}, x\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 54.4

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x}{a}}, x\right) \]

   11. Simplified 54.4%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x}{a}}, x\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(y - a\right) \cdot \frac{x - t}{z}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ t (* (- y a) (/ (- x t) z)))))
   (if (<= z -6.5e+138)
     t_1
     (if (<= z 3.2e+162) (fma (- t x) (/ (- y z) (- a z)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t + ((y - a) * ((x - t) / z));
	double tmp;
	if (z <= -6.5e+138) {
		tmp = t_1;
	} else if (z <= 3.2e+162) {
		tmp = fma((t - x), ((y - z) / (a - z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t + Float64(Float64(y - a) * Float64(Float64(x - t) / z)))
	tmp = 0.0
	if (z <= -6.5e+138)
		tmp = t_1;
	elseif (z <= 3.2e+162)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / Float64(a - z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t + N[(N[(y - a), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.5e+138], t$95$1, If[LessEqual[z, 3.2e+162], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.50000000000000054e138 or 3.2000000000000001e162 < z

   1. Initial program 20.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 52.4

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 52.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      7. div-sub N/A

         \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. associate-/l* N/A

         \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      9. associate-/l* N/A

         \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      14. --lowering--.f64 87.6

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Simplified 87.6%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   if -6.50000000000000054e138 < z < 3.2000000000000001e162

   1. Initial program 82.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 92.1

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 92.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+138}:\\ \;\;\;\;t + \left(y - a\right) \cdot \frac{x - t}{z}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(y - a\right) \cdot \frac{x - t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+72}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-223}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{-28}:\\ \;\;\;\;x - \frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.75e+72)
   t
   (if (<= z -8.5e-223)
     (* t (/ y (- a z)))
     (if (<= z 6.9e-28) (- x (/ (* y x) a)) (* (/ t z) (- z y))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e+72) {
		tmp = t;
	} else if (z <= -8.5e-223) {
		tmp = t * (y / (a - z));
	} else if (z <= 6.9e-28) {
		tmp = x - ((y * x) / a);
	} else {
		tmp = (t / z) * (z - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.75d+72)) then
        tmp = t
    else if (z <= (-8.5d-223)) then
        tmp = t * (y / (a - z))
    else if (z <= 6.9d-28) then
        tmp = x - ((y * x) / a)
    else
        tmp = (t / z) * (z - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e+72) {
		tmp = t;
	} else if (z <= -8.5e-223) {
		tmp = t * (y / (a - z));
	} else if (z <= 6.9e-28) {
		tmp = x - ((y * x) / a);
	} else {
		tmp = (t / z) * (z - y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.75e+72:
		tmp = t
	elif z <= -8.5e-223:
		tmp = t * (y / (a - z))
	elif z <= 6.9e-28:
		tmp = x - ((y * x) / a)
	else:
		tmp = (t / z) * (z - y)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.75e+72)
		tmp = t;
	elseif (z <= -8.5e-223)
		tmp = Float64(t * Float64(y / Float64(a - z)));
	elseif (z <= 6.9e-28)
		tmp = Float64(x - Float64(Float64(y * x) / a));
	else
		tmp = Float64(Float64(t / z) * Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.75e+72)
		tmp = t;
	elseif (z <= -8.5e-223)
		tmp = t * (y / (a - z));
	elseif (z <= 6.9e-28)
		tmp = x - ((y * x) / a);
	else
		tmp = (t / z) * (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.75e+72], t, If[LessEqual[z, -8.5e-223], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.9e-28], N[(x - N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(t / z), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.75000000000000005e72

   1. Initial program 28.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 52.7%

      \[\leadsto \color{blue}{t} \]

   if -1.75000000000000005e72 < z < -8.5000000000000003e-223

   1. Initial program 77.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      4. --lowering--.f64 45.3

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   5. Simplified 45.3%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{t \cdot y}{a - z}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{y}{a - z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \frac{y}{a - z}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto t \cdot \color{blue}{\frac{y}{a - z}} \]

      4. --lowering--.f64 37.4

         \[\leadsto t \cdot \frac{y}{\color{blue}{a - z}} \]

   8. Simplified 37.4%

      \[\leadsto \color{blue}{t \cdot \frac{y}{a - z}} \]

   if -8.5000000000000003e-223 < z < 6.90000000000000001e-28

   1. Initial program 94.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{y - z}{a - z}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y - z}{a - z}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(y - z\right)}{a - z}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - z\right) \cdot x}}{a - z}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \frac{x}{a - z}}\right)\right) + 1 \cdot x \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{x}{a - z}} + 1 \cdot x \]

      10. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(y - z\right)\right)} \cdot \frac{x}{a - z} + 1 \cdot x \]

      11. *-lft-identity N/A

          \[\leadsto \left(-1 \cdot \left(y - z\right)\right) \cdot \frac{x}{a - z} + \color{blue}{x} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - z\right), \frac{x}{a - z}, x\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - z\right)\right)}, \frac{x}{a - z}, x\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right), \frac{x}{a - z}, x\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + y\right)}\right), \frac{x}{a - z}, x\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, \frac{x}{a - z}, x\right) \]

      17. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y}, \frac{x}{a - z}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z} - y, \frac{x}{a - z}, x\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - y}, \frac{x}{a - z}, x\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - y, \color{blue}{\frac{x}{a - z}}, x\right) \]

      21. --lowering--.f64 66.5

          \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{a - z}}, x\right) \]

   5. Simplified 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - y, \frac{x}{a - z}, x\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x \cdot y}{a}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{a}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{x \cdot y}{a}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{x \cdot y}{a}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{x \cdot y}{a}} \]

      5. *-lowering-*.f64 60.7

         \[\leadsto x - \frac{\color{blue}{x \cdot y}}{a} \]

   8. Simplified 60.7%

      \[\leadsto \color{blue}{x - \frac{x \cdot y}{a}} \]

   if 6.90000000000000001e-28 < z

   1. Initial program 51.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      4. --lowering--.f64 45.9

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   5. Simplified 45.9%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{-1 \cdot z}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 38.4

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{-z}} \]

   8. Simplified 38.4%

      \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{-z}} \]
   9. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{-1 \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{z \cdot -1}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{t}{z} \cdot \frac{y - z}{-1}} \]

      4. div-inv N/A

         \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{-1}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{t}{z} \cdot \left(\left(y - z\right) \cdot \color{blue}{-1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{t}{z} \cdot \left(\left(y - z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right) \cdot 1\right)\right)} \]

      8. *-rgt-identity N/A

         \[\leadsto \frac{t}{z} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \frac{t}{z} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)} \]

      11. remove-double-neg N/A

          \[\leadsto \frac{t}{z} \cdot \left(\left(\mathsf{neg}\left(y\right)\right) + \color{blue}{z}\right) \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)} \]

      13. sub-neg N/A

          \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(z - y\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{t}{z} \cdot \left(z - y\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{t}{z}} \cdot \left(z - y\right) \]

      16. --lowering--.f64 44.7

          \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(z - y\right)} \]

   10. Applied egg-rr 44.7%

       \[\leadsto \color{blue}{\frac{t}{z} \cdot \left(z - y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+72}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-223}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{-28}:\\ \;\;\;\;x - \frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{a}, x\right)\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-53}:\\ \;\;\;\;\frac{t}{z} \cdot \left(z - y\right)\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+126}:\\ \;\;\;\;y \cdot \frac{t - x}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.5e+104)
   (fma z (/ x a) x)
   (if (<= a 3.3e-53)
     (* (/ t z) (- z y))
     (if (<= a 2.4e+126) (* y (/ (- t x) a)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.5e+104) {
		tmp = fma(z, (x / a), x);
	} else if (a <= 3.3e-53) {
		tmp = (t / z) * (z - y);
	} else if (a <= 2.4e+126) {
		tmp = y * ((t - x) / a);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.5e+104)
		tmp = fma(z, Float64(x / a), x);
	elseif (a <= 3.3e-53)
		tmp = Float64(Float64(t / z) * Float64(z - y));
	elseif (a <= 2.4e+126)
		tmp = Float64(y * Float64(Float64(t - x) / a));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.5e+104], N[(z * N[(x / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 3.3e-53], N[(N[(t / z), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.4e+126], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.4999999999999998e104

   1. Initial program 58.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{y - z}{a - z}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y - z}{a - z}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(y - z\right)}{a - z}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - z\right) \cdot x}}{a - z}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \frac{x}{a - z}}\right)\right) + 1 \cdot x \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{x}{a - z}} + 1 \cdot x \]

      10. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(y - z\right)\right)} \cdot \frac{x}{a - z} + 1 \cdot x \]

      11. *-lft-identity N/A

          \[\leadsto \left(-1 \cdot \left(y - z\right)\right) \cdot \frac{x}{a - z} + \color{blue}{x} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - z\right), \frac{x}{a - z}, x\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - z\right)\right)}, \frac{x}{a - z}, x\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right), \frac{x}{a - z}, x\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + y\right)}\right), \frac{x}{a - z}, x\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, \frac{x}{a - z}, x\right) \]

      17. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y}, \frac{x}{a - z}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z} - y, \frac{x}{a - z}, x\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - y}, \frac{x}{a - z}, x\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - y, \color{blue}{\frac{x}{a - z}}, x\right) \]

      21. --lowering--.f64 51.4

          \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{a - z}}, x\right) \]

   5. Simplified 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - y, \frac{x}{a - z}, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{z}, \frac{x}{a - z}, x\right) \]
   7. Step-by-step derivation

   8. Simplified 51.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z}, \frac{x}{a - z}, x\right) \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x}{a}}, x\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 49.2

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x}{a}}, x\right) \]

   11. Simplified 49.2%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x}{a}}, x\right) \]

   if -2.4999999999999998e104 < a < 3.30000000000000004e-53

   1. Initial program 70.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      4. --lowering--.f64 51.3

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   5. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{-1 \cdot z}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 42.3

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{-z}} \]

   8. Simplified 42.3%

      \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{-z}} \]
   9. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{-1 \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{z \cdot -1}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{t}{z} \cdot \frac{y - z}{-1}} \]

      4. div-inv N/A

         \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{-1}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{t}{z} \cdot \left(\left(y - z\right) \cdot \color{blue}{-1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{t}{z} \cdot \left(\left(y - z\right) \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right) \cdot 1\right)\right)} \]

      8. *-rgt-identity N/A

         \[\leadsto \frac{t}{z} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \frac{t}{z} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right)\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)\right)} \]

      11. remove-double-neg N/A

          \[\leadsto \frac{t}{z} \cdot \left(\left(\mathsf{neg}\left(y\right)\right) + \color{blue}{z}\right) \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)} \]

      13. sub-neg N/A

          \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(z - y\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{t}{z} \cdot \left(z - y\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{t}{z}} \cdot \left(z - y\right) \]

      16. --lowering--.f64 47.2

          \[\leadsto \frac{t}{z} \cdot \color{blue}{\left(z - y\right)} \]

   10. Applied egg-rr 47.2%

       \[\leadsto \color{blue}{\frac{t}{z} \cdot \left(z - y\right)} \]

   if 3.30000000000000004e-53 < a < 2.40000000000000012e126

   1. Initial program 70.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - x}}{a - z}, y - z, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{t - x}{\color{blue}{a - z}}, y - z, x\right) \]

      8. --lowering--.f64 75.7

         \[\leadsto \mathsf{fma}\left(\frac{t - x}{a - z}, \color{blue}{y - z}, x\right) \]

   4. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{t - x}{\color{blue}{a}}, y - z, x\right) \]
   6. Step-by-step derivation

   7. Simplified 55.8%

      \[\leadsto \mathsf{fma}\left(\frac{t - x}{\color{blue}{a}}, y - z, x\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a} - \frac{x}{a}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{t}{a} - \frac{x}{a}\right)} \]

      2. div-sub N/A

         \[\leadsto y \cdot \color{blue}{\frac{t - x}{a}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{t - x}{a}} \]

      4. --lowering--.f64 38.3

         \[\leadsto y \cdot \frac{\color{blue}{t - x}}{a} \]

   10. Simplified 38.3%

       \[\leadsto \color{blue}{y \cdot \frac{t - x}{a}} \]

   if 2.40000000000000012e126 < a

   1. Initial program 54.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 52.5%

      \[\leadsto \color{blue}{x} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -40:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.85 \cdot 10^{-52}:\\ \;\;\;\;t + \left(y - a\right) \cdot \frac{x - t}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- t x) a) x)))
   (if (<= a -40.0)
     t_1
     (if (<= a 2.85e-52) (+ t (* (- y a) (/ (- x t) z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), ((t - x) / a), x);
	double tmp;
	if (a <= -40.0) {
		tmp = t_1;
	} else if (a <= 2.85e-52) {
		tmp = t + ((y - a) * ((x - t) / z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(Float64(t - x) / a), x)
	tmp = 0.0
	if (a <= -40.0)
		tmp = t_1;
	elseif (a <= 2.85e-52)
		tmp = Float64(t + Float64(Float64(y - a) * Float64(Float64(x - t) / z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -40.0], t$95$1, If[LessEqual[a, 2.85e-52], N[(t + N[(N[(y - a), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -40 or 2.8499999999999999e-52 < a

   1. Initial program 63.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. --lowering--.f64 72.4

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Simplified 72.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

   if -40 < a < 2.8499999999999999e-52

   1. Initial program 68.7%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 76.2

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      7. div-sub N/A

         \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. associate-/l* N/A

         \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      9. associate-/l* N/A

         \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      14. --lowering--.f64 81.6

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Simplified 81.6%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -40:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 2.85 \cdot 10^{-52}:\\ \;\;\;\;t + \left(y - a\right) \cdot \frac{x - t}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+83}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-236}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.2e+83)
   t
   (if (<= z 1.95e-236)
     (* t (/ y (- a z)))
     (if (<= z 1.75e-19) (fma z (/ x a) x) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.2e+83) {
		tmp = t;
	} else if (z <= 1.95e-236) {
		tmp = t * (y / (a - z));
	} else if (z <= 1.75e-19) {
		tmp = fma(z, (x / a), x);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.2e+83)
		tmp = t;
	elseif (z <= 1.95e-236)
		tmp = Float64(t * Float64(y / Float64(a - z)));
	elseif (z <= 1.75e-19)
		tmp = fma(z, Float64(x / a), x);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.2e+83], t, If[LessEqual[z, 1.95e-236], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.75e-19], N[(z * N[(x / a), $MachinePrecision] + x), $MachinePrecision], t]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.1999999999999995e83 or 1.75000000000000008e-19 < z

   1. Initial program 42.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 46.7%

      \[\leadsto \color{blue}{t} \]

   if -7.1999999999999995e83 < z < 1.95e-236

   1. Initial program 82.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      4. --lowering--.f64 43.3

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   5. Simplified 43.3%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{t \cdot y}{a - z}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{y}{a - z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \frac{y}{a - z}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto t \cdot \color{blue}{\frac{y}{a - z}} \]

      4. --lowering--.f64 41.2

         \[\leadsto t \cdot \frac{y}{\color{blue}{a - z}} \]

   8. Simplified 41.2%

      \[\leadsto \color{blue}{t \cdot \frac{y}{a - z}} \]

   if 1.95e-236 < z < 1.75000000000000008e-19

   1. Initial program 97.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{y - z}{a - z}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y - z}{a - z}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(y - z\right)}{a - z}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - z\right) \cdot x}}{a - z}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \frac{x}{a - z}}\right)\right) + 1 \cdot x \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{x}{a - z}} + 1 \cdot x \]

      10. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(y - z\right)\right)} \cdot \frac{x}{a - z} + 1 \cdot x \]

      11. *-lft-identity N/A

          \[\leadsto \left(-1 \cdot \left(y - z\right)\right) \cdot \frac{x}{a - z} + \color{blue}{x} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - z\right), \frac{x}{a - z}, x\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - z\right)\right)}, \frac{x}{a - z}, x\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right), \frac{x}{a - z}, x\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + y\right)}\right), \frac{x}{a - z}, x\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, \frac{x}{a - z}, x\right) \]

      17. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y}, \frac{x}{a - z}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z} - y, \frac{x}{a - z}, x\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - y}, \frac{x}{a - z}, x\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - y, \color{blue}{\frac{x}{a - z}}, x\right) \]

      21. --lowering--.f64 71.9

          \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{a - z}}, x\right) \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - y, \frac{x}{a - z}, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{z}, \frac{x}{a - z}, x\right) \]
   7. Step-by-step derivation

   8. Simplified 52.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z}, \frac{x}{a - z}, x\right) \]

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x}{a}}, x\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 54.4

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x}{a}}, x\right) \]

   11. Simplified 54.4%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x}{a}}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -3.7:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-52}:\\ \;\;\;\;t + y \cdot \frac{x - t}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- t x) a) x)))
   (if (<= a -3.7) t_1 (if (<= a 3.2e-52) (+ t (* y (/ (- x t) z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), ((t - x) / a), x);
	double tmp;
	if (a <= -3.7) {
		tmp = t_1;
	} else if (a <= 3.2e-52) {
		tmp = t + (y * ((x - t) / z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(Float64(t - x) / a), x)
	tmp = 0.0
	if (a <= -3.7)
		tmp = t_1;
	elseif (a <= 3.2e-52)
		tmp = Float64(t + Float64(y * Float64(Float64(x - t) / z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -3.7], t$95$1, If[LessEqual[a, 3.2e-52], N[(t + N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.7000000000000002 or 3.2000000000000001e-52 < a

   1. Initial program 63.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. --lowering--.f64 72.4

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Simplified 72.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

   if -3.7000000000000002 < a < 3.2000000000000001e-52

   1. Initial program 68.7%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 76.2

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      7. div-sub N/A

         \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. associate-/l* N/A

         \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      9. associate-/l* N/A

         \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      14. --lowering--.f64 81.6

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Simplified 81.6%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

   10. Simplified 77.6%

       \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.7:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-52}:\\ \;\;\;\;t + y \cdot \frac{x - t}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+52}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, 0\right)}{z}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-239}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.4e+52)
   t
   (if (<= z -1.16e-127)
     (/ (fma y x 0.0) z)
     (if (<= z 6.6e-239) (* t (/ y a)) (if (<= z 1.7e-19) x t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.4e+52) {
		tmp = t;
	} else if (z <= -1.16e-127) {
		tmp = fma(y, x, 0.0) / z;
	} else if (z <= 6.6e-239) {
		tmp = t * (y / a);
	} else if (z <= 1.7e-19) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.4e+52)
		tmp = t;
	elseif (z <= -1.16e-127)
		tmp = Float64(fma(y, x, 0.0) / z);
	elseif (z <= 6.6e-239)
		tmp = Float64(t * Float64(y / a));
	elseif (z <= 1.7e-19)
		tmp = x;
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.4e+52], t, If[LessEqual[z, -1.16e-127], N[(N[(y * x + 0.0), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 6.6e-239], N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.7e-19], x, t]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.4e52 or 1.7000000000000001e-19 < z

   1. Initial program 42.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 45.0%

      \[\leadsto \color{blue}{t} \]

   if -2.4e52 < z < -1.16e-127

   1. Initial program 80.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{y - z}{a - z}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y - z}{a - z}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(y - z\right)}{a - z}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - z\right) \cdot x}}{a - z}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \frac{x}{a - z}}\right)\right) + 1 \cdot x \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{x}{a - z}} + 1 \cdot x \]

      10. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(y - z\right)\right)} \cdot \frac{x}{a - z} + 1 \cdot x \]

      11. *-lft-identity N/A

          \[\leadsto \left(-1 \cdot \left(y - z\right)\right) \cdot \frac{x}{a - z} + \color{blue}{x} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - z\right), \frac{x}{a - z}, x\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - z\right)\right)}, \frac{x}{a - z}, x\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right), \frac{x}{a - z}, x\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + y\right)}\right), \frac{x}{a - z}, x\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, \frac{x}{a - z}, x\right) \]

      17. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y}, \frac{x}{a - z}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z} - y, \frac{x}{a - z}, x\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - y}, \frac{x}{a - z}, x\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - y, \color{blue}{\frac{x}{a - z}}, x\right) \]

      21. --lowering--.f64 52.3

          \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{a - z}}, x\right) \]

   5. Simplified 52.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - y, \frac{x}{a - z}, x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{-1 \cdot z}}, x\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{\mathsf{neg}\left(z\right)}}, x\right) \]

      2. neg-lowering-neg.f64 28.8

         \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{-z}}, x\right) \]

   8. Simplified 28.8%

      \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{-z}}, x\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(x + -1 \cdot x\right)}{z}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot y + z \cdot \left(x + -1 \cdot x\right)}{z}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot x} + z \cdot \left(x + -1 \cdot x\right)}{z} \]

       3. distribute-rgt1-in N/A

          \[\leadsto \frac{y \cdot x + z \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot x\right)}}{z} \]

       4. metadata-eval N/A

          \[\leadsto \frac{y \cdot x + z \cdot \left(\color{blue}{0} \cdot x\right)}{z} \]

       5. mul0-lft N/A

          \[\leadsto \frac{y \cdot x + z \cdot \color{blue}{0}}{z} \]

       6. mul0-rgt N/A

          \[\leadsto \frac{y \cdot x + \color{blue}{0}}{z} \]

       7. accelerator-lowering-fma.f64 32.7

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, 0\right)}}{z} \]

   11. Simplified 32.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, x, 0\right)}{z}} \]

   if -1.16e-127 < z < 6.5999999999999999e-239

   1. Initial program 88.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      4. --lowering--.f64 45.8

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   5. Simplified 45.8%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot t}}{a} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot t}}{a} \]

      4. --lowering--.f64 37.7

         \[\leadsto \frac{\color{blue}{\left(y - z\right)} \cdot t}{a} \]

   8. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot t}{a}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{y - z}{a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \frac{y - z}{a}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a}} \]

      5. --lowering--.f64 49.1

         \[\leadsto t \cdot \frac{\color{blue}{y - z}}{a} \]

   10. Applied egg-rr 49.1%

       \[\leadsto \color{blue}{t \cdot \frac{y - z}{a}} \]

   11. Taylor expanded in y around inf

       \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 47.0

          \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]

   13. Simplified 47.0%

       \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]

   if 6.5999999999999999e-239 < z < 1.7000000000000001e-19

   1. Initial program 97.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 52.7%

      \[\leadsto \color{blue}{x} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 37.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+51}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-97}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-234}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8e+51)
   t
   (if (<= z -3.8e-97)
     (* x (/ y z))
     (if (<= z 7.5e-234) (* t (/ y a)) (if (<= z 1.66e-19) x t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8e+51) {
		tmp = t;
	} else if (z <= -3.8e-97) {
		tmp = x * (y / z);
	} else if (z <= 7.5e-234) {
		tmp = t * (y / a);
	} else if (z <= 1.66e-19) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8d+51)) then
        tmp = t
    else if (z <= (-3.8d-97)) then
        tmp = x * (y / z)
    else if (z <= 7.5d-234) then
        tmp = t * (y / a)
    else if (z <= 1.66d-19) then
        tmp = x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8e+51) {
		tmp = t;
	} else if (z <= -3.8e-97) {
		tmp = x * (y / z);
	} else if (z <= 7.5e-234) {
		tmp = t * (y / a);
	} else if (z <= 1.66e-19) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8e+51:
		tmp = t
	elif z <= -3.8e-97:
		tmp = x * (y / z)
	elif z <= 7.5e-234:
		tmp = t * (y / a)
	elif z <= 1.66e-19:
		tmp = x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8e+51)
		tmp = t;
	elseif (z <= -3.8e-97)
		tmp = Float64(x * Float64(y / z));
	elseif (z <= 7.5e-234)
		tmp = Float64(t * Float64(y / a));
	elseif (z <= 1.66e-19)
		tmp = x;
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8e+51)
		tmp = t;
	elseif (z <= -3.8e-97)
		tmp = x * (y / z);
	elseif (z <= 7.5e-234)
		tmp = t * (y / a);
	elseif (z <= 1.66e-19)
		tmp = x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8e+51], t, If[LessEqual[z, -3.8e-97], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.5e-234], N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.66e-19], x, t]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8e51 or 1.65999999999999998e-19 < z

   1. Initial program 42.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 45.0%

      \[\leadsto \color{blue}{t} \]

   if -8e51 < z < -3.8000000000000001e-97

   1. Initial program 83.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{y - z}{a - z}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y - z}{a - z}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(y - z\right)}{a - z}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - z\right) \cdot x}}{a - z}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \frac{x}{a - z}}\right)\right) + 1 \cdot x \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{x}{a - z}} + 1 \cdot x \]

      10. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(y - z\right)\right)} \cdot \frac{x}{a - z} + 1 \cdot x \]

      11. *-lft-identity N/A

          \[\leadsto \left(-1 \cdot \left(y - z\right)\right) \cdot \frac{x}{a - z} + \color{blue}{x} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - z\right), \frac{x}{a - z}, x\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - z\right)\right)}, \frac{x}{a - z}, x\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right), \frac{x}{a - z}, x\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + y\right)}\right), \frac{x}{a - z}, x\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, \frac{x}{a - z}, x\right) \]

      17. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y}, \frac{x}{a - z}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z} - y, \frac{x}{a - z}, x\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - y}, \frac{x}{a - z}, x\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - y, \color{blue}{\frac{x}{a - z}}, x\right) \]

      21. --lowering--.f64 52.5

          \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{a - z}}, x\right) \]

   5. Simplified 52.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - y, \frac{x}{a - z}, x\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{-1 \cdot z}}, x\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{\mathsf{neg}\left(z\right)}}, x\right) \]

      2. neg-lowering-neg.f64 30.7

         \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{-z}}, x\right) \]

   8. Simplified 30.7%

      \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{-z}}, x\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{z} \cdot x} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{z} \cdot x} \]

       4. /-lowering-/.f64 32.4

          \[\leadsto \color{blue}{\frac{y}{z}} \cdot x \]

   11. Simplified 32.4%

       \[\leadsto \color{blue}{\frac{y}{z} \cdot x} \]

   if -3.8000000000000001e-97 < z < 7.49999999999999996e-234

   1. Initial program 86.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      4. --lowering--.f64 44.7

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   5. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot t}}{a} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot t}}{a} \]

      4. --lowering--.f64 35.8

         \[\leadsto \frac{\color{blue}{\left(y - z\right)} \cdot t}{a} \]

   8. Simplified 35.8%

      \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot t}{a}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{y - z}{a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \frac{y - z}{a}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a}} \]

      5. --lowering--.f64 45.9

         \[\leadsto t \cdot \frac{\color{blue}{y - z}}{a} \]

   10. Applied egg-rr 45.9%

       \[\leadsto \color{blue}{t \cdot \frac{y - z}{a}} \]

   11. Taylor expanded in y around inf

       \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 44.1

          \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]

   13. Simplified 44.1%

       \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]

   if 7.49999999999999996e-234 < z < 1.65999999999999998e-19

   1. Initial program 97.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 52.7%

      \[\leadsto \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+51}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-97}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-234}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{a}, y - z, x\right)\\ \mathbf{if}\;a \leq -2400:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-52}:\\ \;\;\;\;t + y \cdot \frac{x - t}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ t a) (- y z) x)))
   (if (<= a -2400.0) t_1 (if (<= a 3.2e-52) (+ t (* y (/ (- x t) z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t / a), (y - z), x);
	double tmp;
	if (a <= -2400.0) {
		tmp = t_1;
	} else if (a <= 3.2e-52) {
		tmp = t + (y * ((x - t) / z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t / a), Float64(y - z), x)
	tmp = 0.0
	if (a <= -2400.0)
		tmp = t_1;
	elseif (a <= 3.2e-52)
		tmp = Float64(t + Float64(y * Float64(Float64(x - t) / z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / a), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2400.0], t$95$1, If[LessEqual[a, 3.2e-52], N[(t + N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2400 or 3.2000000000000001e-52 < a

   1. Initial program 63.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - x}}{a - z}, y - z, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{t - x}{\color{blue}{a - z}}, y - z, x\right) \]

      8. --lowering--.f64 86.7

         \[\leadsto \mathsf{fma}\left(\frac{t - x}{a - z}, \color{blue}{y - z}, x\right) \]

   4. Applied egg-rr 86.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{t - x}{\color{blue}{a}}, y - z, x\right) \]
   6. Step-by-step derivation

   7. Simplified 72.4%

      \[\leadsto \mathsf{fma}\left(\frac{t - x}{\color{blue}{a}}, y - z, x\right) \]

   8. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t}}{a}, y - z, x\right) \]
   9. Step-by-step derivation

   10. Simplified 65.1%

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t}}{a}, y - z, x\right) \]

   if -2400 < a < 3.2000000000000001e-52

   1. Initial program 68.7%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 76.2

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      7. div-sub N/A

         \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. associate-/l* N/A

         \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      9. associate-/l* N/A

         \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      14. --lowering--.f64 81.6

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Simplified 81.6%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{y} \]
   9. Step-by-step derivation

   10. Simplified 77.6%

       \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2400:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y - z, x\right)\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-52}:\\ \;\;\;\;t + y \cdot \frac{x - t}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{a}, y - z, x\right)\\ \mathbf{if}\;a \leq -920:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ t a) (- y z) x)))
   (if (<= a -920.0) t_1 (if (<= a 3.2e-52) (fma (- x t) (/ y z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t / a), (y - z), x);
	double tmp;
	if (a <= -920.0) {
		tmp = t_1;
	} else if (a <= 3.2e-52) {
		tmp = fma((x - t), (y / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t / a), Float64(y - z), x)
	tmp = 0.0
	if (a <= -920.0)
		tmp = t_1;
	elseif (a <= 3.2e-52)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / a), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -920.0], t$95$1, If[LessEqual[a, 3.2e-52], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -920 or 3.2000000000000001e-52 < a

   1. Initial program 63.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - x}}{a - z}, y - z, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{t - x}{\color{blue}{a - z}}, y - z, x\right) \]

      8. --lowering--.f64 86.7

         \[\leadsto \mathsf{fma}\left(\frac{t - x}{a - z}, \color{blue}{y - z}, x\right) \]

   4. Applied egg-rr 86.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{t - x}{\color{blue}{a}}, y - z, x\right) \]
   6. Step-by-step derivation

   7. Simplified 72.4%

      \[\leadsto \mathsf{fma}\left(\frac{t - x}{\color{blue}{a}}, y - z, x\right) \]

   8. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t}}{a}, y - z, x\right) \]
   9. Step-by-step derivation

   10. Simplified 65.1%

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t}}{a}, y - z, x\right) \]

   if -920 < a < 3.2000000000000001e-52

   1. Initial program 68.7%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 76.2

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      7. div-sub N/A

         \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. associate-/l* N/A

         \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      9. associate-/l* N/A

         \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      14. --lowering--.f64 81.6

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Simplified 81.6%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right)}{z}} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{t + \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right)\right) + t} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(t - x\right) \cdot y}}{z}\right)\right) + t \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(t - x\right) \cdot \frac{y}{z}}\right)\right) + t \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \frac{y}{z}} + t \]

      6. neg-mul-1 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t - x\right)\right)} \cdot \frac{y}{z} + t \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(t - x\right), \frac{y}{z}, t\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(t - x\right)\right)}, \frac{y}{z}, t\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(x\right)\right)\right)}\right), \frac{y}{z}, t\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\left(t + \color{blue}{-1 \cdot x}\right)\right), \frac{y}{z}, t\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot x + t\right)}\right), \frac{y}{z}, t\right) \]

      12. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(t\right)\right)}, \frac{y}{z}, t\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right) + \left(\mathsf{neg}\left(t\right)\right), \frac{y}{z}, t\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x} + \left(\mathsf{neg}\left(t\right)\right), \frac{y}{z}, t\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x - t}, \frac{y}{z}, t\right) \]

      16. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x - t}, \frac{y}{z}, t\right) \]

      17. /-lowering-/.f64 76.2

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y}{z}}, t\right) \]

   10. Simplified 76.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 69.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -330:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -330.0)
   (fma y (/ (- t x) a) x)
   (if (<= a 3.2e-52) (fma (- x t) (/ y z) t) (fma (- t x) (/ y a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -330.0) {
		tmp = fma(y, ((t - x) / a), x);
	} else if (a <= 3.2e-52) {
		tmp = fma((x - t), (y / z), t);
	} else {
		tmp = fma((t - x), (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -330.0)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	elseif (a <= 3.2e-52)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	else
		tmp = fma(Float64(t - x), Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -330.0], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 3.2e-52], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -330

   1. Initial program 65.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{t - x}{a}} + x \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t - x}{a}}, x\right) \]

      5. --lowering--.f64 66.5

         \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Simplified 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)} \]

   if -330 < a < 3.2000000000000001e-52

   1. Initial program 68.7%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 76.2

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      7. div-sub N/A

         \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. associate-/l* N/A

         \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      9. associate-/l* N/A

         \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      14. --lowering--.f64 81.6

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Simplified 81.6%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right)}{z}} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{t + \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right)}{z}\right)\right) + t} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(t - x\right) \cdot y}}{z}\right)\right) + t \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(t - x\right) \cdot \frac{y}{z}}\right)\right) + t \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \frac{y}{z}} + t \]

      6. neg-mul-1 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t - x\right)\right)} \cdot \frac{y}{z} + t \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(t - x\right), \frac{y}{z}, t\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(t - x\right)\right)}, \frac{y}{z}, t\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(t + \left(\mathsf{neg}\left(x\right)\right)\right)}\right), \frac{y}{z}, t\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\left(t + \color{blue}{-1 \cdot x}\right)\right), \frac{y}{z}, t\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot x + t\right)}\right), \frac{y}{z}, t\right) \]

      12. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(t\right)\right)}, \frac{y}{z}, t\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right) + \left(\mathsf{neg}\left(t\right)\right), \frac{y}{z}, t\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x} + \left(\mathsf{neg}\left(t\right)\right), \frac{y}{z}, t\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x - t}, \frac{y}{z}, t\right) \]

      16. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x - t}, \frac{y}{z}, t\right) \]

      17. /-lowering-/.f64 76.2

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y}{z}}, t\right) \]

   10. Simplified 76.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)} \]

   if 3.2000000000000001e-52 < a

   1. Initial program 61.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 88.3

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 88.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y}{a}}, x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 60.6

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y}{a}}, x\right) \]

   7. Simplified 60.6%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y}{a}}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 63.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{a}{z}, t\right)\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ a z) t)))
   (if (<= z -2.8e+72) t_1 (if (<= z 5e+63) (fma (- t x) (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), (a / z), t);
	double tmp;
	if (z <= -2.8e+72) {
		tmp = t_1;
	} else if (z <= 5e+63) {
		tmp = fma((t - x), (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(a / z), t)
	tmp = 0.0
	if (z <= -2.8e+72)
		tmp = t_1;
	elseif (z <= 5e+63)
		tmp = fma(Float64(t - x), Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(a / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -2.8e+72], t$95$1, If[LessEqual[z, 5e+63], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7999999999999999e72 or 5.00000000000000011e63 < z

   1. Initial program 33.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 65.6

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      7. div-sub N/A

         \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. associate-/l* N/A

         \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      9. associate-/l* N/A

         \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      14. --lowering--.f64 82.3

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Simplified 82.3%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{t - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{t + \left(\mathsf{neg}\left(-1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto t + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot \left(t - x\right)}{z}\right)\right)}\right)\right) \]

      3. remove-double-neg N/A

         \[\leadsto t + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot \left(t - x\right)}{z} + t} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot a}}{z} + t \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{a}{z}} + t \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{a}{z}, t\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{a}{z}, t\right) \]

      9. /-lowering-/.f64 61.4

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{a}{z}}, t\right) \]

   10. Simplified 61.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{a}{z}, t\right)} \]

   if -2.7999999999999999e72 < z < 5.00000000000000011e63

   1. Initial program 87.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 92.4

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 92.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y}{a}}, x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 64.9

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y}{a}}, x\right) \]

   7. Simplified 64.9%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y}{a}}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 62.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{a}{z}, t\right)\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ a z) t)))
   (if (<= z -2.2e+77) t_1 (if (<= z 9.6e+63) (fma y (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), (a / z), t);
	double tmp;
	if (z <= -2.2e+77) {
		tmp = t_1;
	} else if (z <= 9.6e+63) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(a / z), t)
	tmp = 0.0
	if (z <= -2.2e+77)
		tmp = t_1;
	elseif (z <= 9.6e+63)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(a / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -2.2e+77], t$95$1, If[LessEqual[z, 9.6e+63], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2e77 or 9.6e63 < z

   1. Initial program 33.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{y - z}{a - z}, x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y - z}{a - z}}, x\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(t - x, \frac{\color{blue}{y - z}}{a - z}, x\right) \]

      8. --lowering--.f64 65.6

         \[\leadsto \mathsf{fma}\left(t - x, \frac{y - z}{\color{blue}{a - z}}, x\right) \]

   4. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      6. --lowering--.f64 N/A

         \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      7. div-sub N/A

         \[\leadsto t - \color{blue}{\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      8. associate-/l* N/A

         \[\leadsto t - \left(\color{blue}{y \cdot \frac{t - x}{z}} - \frac{a \cdot \left(t - x\right)}{z}\right) \]

      9. associate-/l* N/A

         \[\leadsto t - \left(y \cdot \frac{t - x}{z} - \color{blue}{a \cdot \frac{t - x}{z}}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z} \cdot \left(y - a\right)} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{t - x}{z}} \cdot \left(y - a\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto t - \frac{\color{blue}{t - x}}{z} \cdot \left(y - a\right) \]

      14. --lowering--.f64 82.3

          \[\leadsto t - \frac{t - x}{z} \cdot \color{blue}{\left(y - a\right)} \]

   7. Simplified 82.3%

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{t - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{t + \left(\mathsf{neg}\left(-1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto t + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot \left(t - x\right)}{z}\right)\right)}\right)\right) \]

      3. remove-double-neg N/A

         \[\leadsto t + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot \left(t - x\right)}{z} + t} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t - x\right) \cdot a}}{z} + t \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \frac{a}{z}} + t \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{a}{z}, t\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t - x}, \frac{a}{z}, t\right) \]

      9. /-lowering-/.f64 61.4

         \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{a}{z}}, t\right) \]

   10. Simplified 61.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{a}{z}, t\right)} \]

   if -2.2e77 < z < 9.6e63

   1. Initial program 87.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{t - x}{a}} + x \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t - x}{a}}, x\right) \]

      5. --lowering--.f64 63.1

         \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Simplified 63.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 59.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+74}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9e+74) t (if (<= z 8.5e+63) (fma y (/ (- t x) a) x) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9e+74) {
		tmp = t;
	} else if (z <= 8.5e+63) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9e+74)
		tmp = t;
	elseif (z <= 8.5e+63)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9e+74], t, If[LessEqual[z, 8.5e+63], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if z < -8.9999999999999999e74 or 8.5000000000000004e63 < z

   1. Initial program 33.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 50.1%

      \[\leadsto \color{blue}{t} \]

   if -8.9999999999999999e74 < z < 8.5000000000000004e63

   1. Initial program 87.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{t - x}{a}} + x \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t - x}{a}}, x\right) \]

      5. --lowering--.f64 63.1

         \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Simplified 63.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 36.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+72}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-233}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.75e+72)
   t
   (if (<= z 7e-233) (* t (/ y a)) (if (<= z 2e-19) x t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e+72) {
		tmp = t;
	} else if (z <= 7e-233) {
		tmp = t * (y / a);
	} else if (z <= 2e-19) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.75d+72)) then
        tmp = t
    else if (z <= 7d-233) then
        tmp = t * (y / a)
    else if (z <= 2d-19) then
        tmp = x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e+72) {
		tmp = t;
	} else if (z <= 7e-233) {
		tmp = t * (y / a);
	} else if (z <= 2e-19) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.75e+72:
		tmp = t
	elif z <= 7e-233:
		tmp = t * (y / a)
	elif z <= 2e-19:
		tmp = x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.75e+72)
		tmp = t;
	elseif (z <= 7e-233)
		tmp = Float64(t * Float64(y / a));
	elseif (z <= 2e-19)
		tmp = x;
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.75e+72)
		tmp = t;
	elseif (z <= 7e-233)
		tmp = t * (y / a);
	elseif (z <= 2e-19)
		tmp = x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.75e+72], t, If[LessEqual[z, 7e-233], N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e-19], x, t]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.75000000000000005e72 or 2e-19 < z

   1. Initial program 42.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 46.7%

      \[\leadsto \color{blue}{t} \]

   if -1.75000000000000005e72 < z < 6.99999999999999982e-233

   1. Initial program 82.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      4. --lowering--.f64 43.3

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   5. Simplified 43.3%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot t}}{a} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot t}}{a} \]

      4. --lowering--.f64 30.4

         \[\leadsto \frac{\color{blue}{\left(y - z\right)} \cdot t}{a} \]

   8. Simplified 30.4%

      \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot t}{a}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{t \cdot \frac{y - z}{a}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \frac{y - z}{a}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a}} \]

      5. --lowering--.f64 37.5

         \[\leadsto t \cdot \frac{\color{blue}{y - z}}{a} \]

   10. Applied egg-rr 37.5%

       \[\leadsto \color{blue}{t \cdot \frac{y - z}{a}} \]

   11. Taylor expanded in y around inf

       \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 32.6

          \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]

   13. Simplified 32.6%

       \[\leadsto t \cdot \color{blue}{\frac{y}{a}} \]

   if 6.99999999999999982e-233 < z < 2e-19

   1. Initial program 97.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 52.7%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 38.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+72}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.6e+72) t (if (<= z 1.55e-19) x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.6e+72) {
		tmp = t;
	} else if (z <= 1.55e-19) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-9.6d+72)) then
        tmp = t
    else if (z <= 1.55d-19) then
        tmp = x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.6e+72) {
		tmp = t;
	} else if (z <= 1.55e-19) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -9.6e+72:
		tmp = t
	elif z <= 1.55e-19:
		tmp = x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.6e+72)
		tmp = t;
	elseif (z <= 1.55e-19)
		tmp = x;
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -9.6e+72)
		tmp = t;
	elseif (z <= 1.55e-19)
		tmp = x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.6e+72], t, If[LessEqual[z, 1.55e-19], x, t]]

Derivation

1. Split input into 2 regimes

2. if z < -9.6000000000000004e72 or 1.5499999999999999e-19 < z

   1. Initial program 42.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 46.7%

      \[\leadsto \color{blue}{t} \]

   if -9.6000000000000004e72 < z < 1.5499999999999999e-19

   1. Initial program 87.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 32.3%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 25.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 t)

C

double code(double x, double y, double z, double t, double a) {
	return t;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return t;
}

Python

def code(x, y, z, t, a):
	return t

Julia

function code(x, y, z, t, a)
	return t
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = t;
end

Wolfram

code[x_, y_, z_, t_, a_] := t

Derivation

1. Initial program 66.2%

   \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{t} \]
4. Step-by-step derivation

5. Simplified 25.6%

   \[\leadsto \color{blue}{t} \]
6. Add Preprocessing

Alternative 21: 2.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 66.2%

   \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{y - z}{a - z}\right) \cdot x + 1 \cdot x} \]

   3. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} \cdot x + 1 \cdot x \]

   4. distribute-lft-neg-out N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z} \cdot x\right)\right)} + 1 \cdot x \]

   5. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y - z}{a - z}}\right)\right) + 1 \cdot x \]

   6. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(y - z\right)}{a - z}}\right)\right) + 1 \cdot x \]

   7. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - z\right) \cdot x}}{a - z}\right)\right) + 1 \cdot x \]

   8. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \frac{x}{a - z}}\right)\right) + 1 \cdot x \]

   9. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{x}{a - z}} + 1 \cdot x \]

   10. mul-1-neg N/A

       \[\leadsto \color{blue}{\left(-1 \cdot \left(y - z\right)\right)} \cdot \frac{x}{a - z} + 1 \cdot x \]

   11. *-lft-identity N/A

       \[\leadsto \left(-1 \cdot \left(y - z\right)\right) \cdot \frac{x}{a - z} + \color{blue}{x} \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - z\right), \frac{x}{a - z}, x\right)} \]

   13. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - z\right)\right)}, \frac{x}{a - z}, x\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right), \frac{x}{a - z}, x\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + y\right)}\right), \frac{x}{a - z}, x\right) \]

   16. distribute-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, \frac{x}{a - z}, x\right) \]

   17. unsub-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y}, \frac{x}{a - z}, x\right) \]

   18. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z} - y, \frac{x}{a - z}, x\right) \]

   19. --lowering--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z - y}, \frac{x}{a - z}, x\right) \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(z - y, \color{blue}{\frac{x}{a - z}}, x\right) \]

   21. --lowering--.f64 38.9

       \[\leadsto \mathsf{fma}\left(z - y, \frac{x}{\color{blue}{a - z}}, x\right) \]

5. Simplified 38.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z - y, \frac{x}{a - z}, x\right)} \]

6. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x + -1 \cdot x} \]
7. Step-by-step derivation

   1. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(-1 + 1\right) \cdot x} \]

   2. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot x \]

   3. mul0-lft 2.7

      \[\leadsto \color{blue}{0} \]

8. Simplified 2.7%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Developer Target 1: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* (/ y z) (- t x)))))
   (if (< z -1.2536131056095036e+188)
     t_1
     (if (< z 4.446702369113811e+64)
       (+ x (/ (- y z) (/ (- a z) (- t x))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t - ((y / z) * (t - x))
    if (z < (-1.2536131056095036d+188)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x + ((y - z) / ((a - z) / (t - x)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - ((y / z) * (t - x))
	tmp = 0
	if z < -1.2536131056095036e+188:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x + ((y - z) / ((a - z) / (t - x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(y / z) * Float64(t - x)))
	tmp = 0.0
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x + Float64(Float64(y - z) / Float64(Float64(a - z) / Float64(t - x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - ((y / z) * (t - x));
	tmp = 0.0;
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(N[(y / z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.2536131056095036e+188], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x + N[(N[(y - z), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024205 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -125361310560950360000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- t (* (/ y z) (- t x))) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x))))))

  (+ x (/ (* (- y z) (- t x)) (- a z))))