Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 80.1% → 94.6%

Time: 12.9s

Alternatives: 14

Speedup: 0.3×

Specification

Math

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

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(y - z) * Float64(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[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 80.1% accurate, 1.0× speedup

Math

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

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(y - z) * Float64(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[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-247}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) (- a z)) (- t x) x))
        (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 -2e-247)
     t_1
     (if (<= t_2 0.0) (fma (/ (- t x) z) (- a y) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / (a - z)), (t - x), x);
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -2e-247) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma(((t - x) / z), (a - y), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / Float64(a - z)), Float64(t - x), x)
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= -2e-247)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(Float64(t - x) / z), Float64(a - y), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-247], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2e-247 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 88.1%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 91.9

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

   4. Applied egg-rr 91.9%

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

   if -2e-247 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 4.3%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

   5. Simplified 98.9%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -2 \cdot 10^{-247}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-196}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;z \leq 106000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ (- t x) z) t)))
   (if (<= z -8.5e+105)
     t_1
     (if (<= z 3.2e-196)
       (fma (/ y a) t x)
       (if (<= z 106000000000.0) (fma (/ y a) (- x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, ((t - x) / z), t);
	double tmp;
	if (z <= -8.5e+105) {
		tmp = t_1;
	} else if (z <= 3.2e-196) {
		tmp = fma((y / a), t, x);
	} else if (z <= 106000000000.0) {
		tmp = fma((y / a), -x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(Float64(t - x) / z), t)
	tmp = 0.0
	if (z <= -8.5e+105)
		tmp = t_1;
	elseif (z <= 3.2e-196)
		tmp = fma(Float64(y / a), t, x);
	elseif (z <= 106000000000.0)
		tmp = fma(Float64(y / a), Float64(-x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -8.5e+105], t$95$1, If[LessEqual[z, 3.2e-196], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 106000000000.0], N[(N[(y / a), $MachinePrecision] * (-x) + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.49999999999999986e105 or 1.06e11 < z

   1. Initial program 62.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 50.4

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

   5. Simplified 50.4%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 64.6

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

   8. Simplified 64.6%

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

   if -8.49999999999999986e105 < z < 3.2e-196

   1. Initial program 91.9%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 94.3

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

   4. Applied egg-rr 94.3%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 72.7

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

   7. Simplified 72.7%

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

   8. Taylor expanded in t around inf

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

   10. Simplified 64.0%

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

   if 3.2e-196 < z < 1.06e11

   1. Initial program 88.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 90.1

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

   4. Applied egg-rr 90.1%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 62.0

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

   7. Simplified 62.0%

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

   8. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 57.7

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

   10. Simplified 57.7%

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

Alternative 3: 54.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{x}{-z}, t\right)\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-196}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ x (- z)) t)))
   (if (<= z -7.5e+99)
     t_1
     (if (<= z 4e-196)
       (fma (/ y a) t x)
       (if (<= z 1.1e+15) (fma (/ y a) (- x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, (x / -z), t);
	double tmp;
	if (z <= -7.5e+99) {
		tmp = t_1;
	} else if (z <= 4e-196) {
		tmp = fma((y / a), t, x);
	} else if (z <= 1.1e+15) {
		tmp = fma((y / a), -x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(x / Float64(-z)), t)
	tmp = 0.0
	if (z <= -7.5e+99)
		tmp = t_1;
	elseif (z <= 4e-196)
		tmp = fma(Float64(y / a), t, x);
	elseif (z <= 1.1e+15)
		tmp = fma(Float64(y / a), Float64(-x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(x / (-z)), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -7.5e+99], t$95$1, If[LessEqual[z, 4e-196], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 1.1e+15], N[(N[(y / a), $MachinePrecision] * (-x) + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.49999999999999963e99 or 1.1e15 < z

   1. Initial program 62.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 50.4

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

   5. Simplified 50.4%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 64.6

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

   8. Simplified 64.6%

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

   9. Taylor expanded in t around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 64.5

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

   11. Simplified 64.5%

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

   if -7.49999999999999963e99 < z < 4.0000000000000002e-196

   1. Initial program 91.9%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 94.3

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

   4. Applied egg-rr 94.3%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 72.7

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

   7. Simplified 72.7%

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

   8. Taylor expanded in t around inf

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

   10. Simplified 64.0%

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

   if 4.0000000000000002e-196 < z < 1.1e15

   1. Initial program 88.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 90.1

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

   4. Applied egg-rr 90.1%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 62.0

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

   7. Simplified 62.0%

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

   8. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 57.7

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

   10. Simplified 57.7%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{x}{-z}, t\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-196}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{x}{-z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-90}:\\ \;\;\;\;t + \frac{\left(t - x\right) \cdot \left(a - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- t x) a) x)))
   (if (<= a -8.2e-5)
     t_1
     (if (<= a 6.2e-90) (+ t (/ (* (- t x) (- a y)) 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 <= -8.2e-5) {
		tmp = t_1;
	} else if (a <= 6.2e-90) {
		tmp = t + (((t - x) * (a - y)) / 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 <= -8.2e-5)
		tmp = t_1;
	elseif (a <= 6.2e-90)
		tmp = Float64(t + Float64(Float64(Float64(t - x) * Float64(a - y)) / 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, -8.2e-5], t$95$1, If[LessEqual[a, 6.2e-90], N[(t + N[(N[(N[(t - x), $MachinePrecision] * N[(a - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.20000000000000009e-5 or 6.2000000000000003e-90 < a

   1. Initial program 88.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 75.9

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

   5. Simplified 75.9%

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

   if -8.20000000000000009e-5 < a < 6.2000000000000003e-90

   1. Initial program 71.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 78.0

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

   4. Applied egg-rr 78.0%

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

   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. /-lowering-/.f64 N/A

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

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

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

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

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

      10. --lowering--.f64 N/A

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

      11. --lowering--.f64 79.6

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

   7. Simplified 79.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-90}:\\ \;\;\;\;t + \frac{\left(t - x\right) \cdot \left(a - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -0.000108:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- t x) a) x)))
   (if (<= a -0.000108)
     t_1
     (if (<= a 1.1e-89) (fma (/ (- t x) z) (- a y) t) 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 <= -0.000108) {
		tmp = t_1;
	} else if (a <= 1.1e-89) {
		tmp = fma(((t - x) / z), (a - y), t);
	} 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 <= -0.000108)
		tmp = t_1;
	elseif (a <= 1.1e-89)
		tmp = fma(Float64(Float64(t - x) / z), Float64(a - y), t);
	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, -0.000108], t$95$1, If[LessEqual[a, 1.1e-89], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.08e-4 or 1.10000000000000006e-89 < a

   1. Initial program 88.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 75.9

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

   5. Simplified 75.9%

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

   if -1.08e-4 < a < 1.10000000000000006e-89

   1. Initial program 71.4%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

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

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

   5. Simplified 80.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000108:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.46 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ (- t x) z) t)))
   (if (<= z -5e+103)
     t_1
     (if (<= z 1.46e+94) (fma (- y z) (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, ((t - x) / z), t);
	double tmp;
	if (z <= -5e+103) {
		tmp = t_1;
	} else if (z <= 1.46e+94) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(Float64(t - x) / z), t)
	tmp = 0.0
	if (z <= -5e+103)
		tmp = t_1;
	elseif (z <= 1.46e+94)
		tmp = fma(Float64(y - z), 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[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -5e+103], t$95$1, If[LessEqual[z, 1.46e+94], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5e103 or 1.46000000000000005e94 < z

   1. Initial program 57.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 51.9

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

   5. Simplified 51.9%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 71.5

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

   8. Simplified 71.5%

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

   if -5e103 < z < 1.46000000000000005e94

   1. Initial program 90.1%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 70.5

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

   5. Simplified 70.5%

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

Alternative 7: 63.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ (- t x) z) t)))
   (if (<= z -2.3e+102) t_1 (if (<= z 2.35e+94) (fma (/ y a) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, ((t - x) / z), t);
	double tmp;
	if (z <= -2.3e+102) {
		tmp = t_1;
	} else if (z <= 2.35e+94) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(Float64(t - x) / z), t)
	tmp = 0.0
	if (z <= -2.3e+102)
		tmp = t_1;
	elseif (z <= 2.35e+94)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -2.3e+102], t$95$1, If[LessEqual[z, 2.35e+94], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2999999999999999e102 or 2.35000000000000008e94 < z

   1. Initial program 57.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 51.9

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

   5. Simplified 51.9%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 71.5

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

   8. Simplified 71.5%

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

   if -2.2999999999999999e102 < z < 2.35000000000000008e94

   1. Initial program 90.1%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 92.0

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

   4. Applied egg-rr 92.0%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 66.9

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

   7. Simplified 66.9%

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

Alternative 8: 62.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ (- t x) z) t)))
   (if (<= z -1.3e+100) t_1 (if (<= z 1.35e+94) (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(a, ((t - x) / z), t);
	double tmp;
	if (z <= -1.3e+100) {
		tmp = t_1;
	} else if (z <= 1.35e+94) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(Float64(t - x) / z), t)
	tmp = 0.0
	if (z <= -1.3e+100)
		tmp = t_1;
	elseif (z <= 1.35e+94)
		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[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.3e+100], t$95$1, If[LessEqual[z, 1.35e+94], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3000000000000001e100 or 1.3500000000000001e94 < z

   1. Initial program 57.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 51.9

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

   5. Simplified 51.9%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 71.5

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

   8. Simplified 71.5%

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

   if -1.3000000000000001e100 < z < 1.3500000000000001e94

   1. Initial program 90.1%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 66.7

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

   5. Simplified 66.7%

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

Alternative 9: 56.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{x}{-z}, t\right)\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ x (- z)) t)))
   (if (<= z -1.65e+100) t_1 (if (<= z 2e+94) (fma (/ y a) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, (x / -z), t);
	double tmp;
	if (z <= -1.65e+100) {
		tmp = t_1;
	} else if (z <= 2e+94) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(x / Float64(-z)), t)
	tmp = 0.0
	if (z <= -1.65e+100)
		tmp = t_1;
	elseif (z <= 2e+94)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(x / (-z)), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.65e+100], t$95$1, If[LessEqual[z, 2e+94], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6500000000000001e100 or 2e94 < z

   1. Initial program 57.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 51.9

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

   5. Simplified 51.9%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 71.5

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

   8. Simplified 71.5%

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

   9. Taylor expanded in t around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 71.3

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

   11. Simplified 71.3%

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

   if -1.6500000000000001e100 < z < 2e94

   1. Initial program 90.1%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 92.0

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

   4. Applied egg-rr 92.0%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 66.9

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

   7. Simplified 66.9%

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

   8. Taylor expanded in t around inf

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

   10. Simplified 56.7%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{x}{-z}, t\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{x}{-z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ t z) t)))
   (if (<= z -2.4e+108) t_1 (if (<= z 3e+94) (fma (/ y a) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, (t / z), t);
	double tmp;
	if (z <= -2.4e+108) {
		tmp = t_1;
	} else if (z <= 3e+94) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(t / z), t)
	tmp = 0.0
	if (z <= -2.4e+108)
		tmp = t_1;
	elseif (z <= 3e+94)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(t / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -2.4e+108], t$95$1, If[LessEqual[z, 3e+94], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.40000000000000019e108 or 3.0000000000000001e94 < z

   1. Initial program 58.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 52.7

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

   5. Simplified 52.7%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 71.1

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

   8. Simplified 71.1%

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

   9. Taylor expanded in t around inf

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

   11. Simplified 63.4%

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

   if -2.40000000000000019e108 < z < 3.0000000000000001e94

   1. Initial program 89.6%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 91.6

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

   4. Applied egg-rr 91.6%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 66.5

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

   7. Simplified 66.5%

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

   8. Taylor expanded in t around inf

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

   10. Simplified 56.5%

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

Alternative 11: 39.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{y}{a}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+201}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (/ y a))))
   (if (<= y -1.6e+168) t_1 (if (<= y 1.15e+201) (+ x t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (y / a);
	double tmp;
	if (y <= -1.6e+168) {
		tmp = t_1;
	} else if (y <= 1.15e+201) {
		tmp = x + t;
	} 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 / a)
    if (y <= (-1.6d+168)) then
        tmp = t_1
    else if (y <= 1.15d+201) then
        tmp = x + t
    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 / a);
	double tmp;
	if (y <= -1.6e+168) {
		tmp = t_1;
	} else if (y <= 1.15e+201) {
		tmp = x + t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t * (y / a)
	tmp = 0
	if y <= -1.6e+168:
		tmp = t_1
	elif y <= 1.15e+201:
		tmp = x + t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(y / a))
	tmp = 0.0
	if (y <= -1.6e+168)
		tmp = t_1;
	elseif (y <= 1.15e+201)
		tmp = Float64(x + t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (y / a);
	tmp = 0.0;
	if (y <= -1.6e+168)
		tmp = t_1;
	elseif (y <= 1.15e+201)
		tmp = x + t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e+168], t$95$1, If[LessEqual[y, 1.15e+201], N[(x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6000000000000001e168 or 1.1500000000000001e201 < y

   1. Initial program 90.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 34.9

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

   5. Simplified 34.9%

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

   6. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

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

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

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

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

      4. --lowering--.f64 42.8

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

   8. Simplified 42.8%

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

   9. Taylor expanded in y around inf

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

       1. /-lowering-/.f64 43.0

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

   11. Simplified 43.0%

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

   if -1.6000000000000001e168 < y < 1.1500000000000001e201

   1. Initial program 78.8%

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

   3. Taylor expanded in z around inf

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

      1. --lowering--.f64 24.5

         \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   5. Simplified 24.5%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{t} \]
   7. Step-by-step derivation

   8. Simplified 49.4%

      \[\leadsto x + \color{blue}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 38.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+107}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+91}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.6e+107) t (if (<= z 5.7e-129) x (if (<= z 1.22e+91) (+ x t) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e+107) {
		tmp = t;
	} else if (z <= 5.7e-129) {
		tmp = x;
	} else if (z <= 1.22e+91) {
		tmp = x + t;
	} 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 <= (-1.6d+107)) then
        tmp = t
    else if (z <= 5.7d-129) then
        tmp = x
    else if (z <= 1.22d+91) then
        tmp = x + t
    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 <= -1.6e+107) {
		tmp = t;
	} else if (z <= 5.7e-129) {
		tmp = x;
	} else if (z <= 1.22e+91) {
		tmp = x + t;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.6e+107:
		tmp = t
	elif z <= 5.7e-129:
		tmp = x
	elif z <= 1.22e+91:
		tmp = x + t
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.6e+107)
		tmp = t;
	elseif (z <= 5.7e-129)
		tmp = x;
	elseif (z <= 1.22e+91)
		tmp = Float64(x + t);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.6e+107)
		tmp = t;
	elseif (z <= 5.7e-129)
		tmp = x;
	elseif (z <= 1.22e+91)
		tmp = x + t;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.6e+107], t, If[LessEqual[z, 5.7e-129], x, If[LessEqual[z, 1.22e+91], N[(x + t), $MachinePrecision], t]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.60000000000000015e107 or 1.2199999999999999e91 < z

   1. Initial program 58.8%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 62.4%

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

   if -1.60000000000000015e107 < z < 5.7000000000000001e-129

   1. Initial program 92.3%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 42.6%

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

   if 5.7000000000000001e-129 < z < 1.2199999999999999e91

   1. Initial program 82.6%

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

   3. Taylor expanded in z around inf

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

      1. --lowering--.f64 17.5

         \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   5. Simplified 17.5%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto x + \color{blue}{t} \]
   7. Step-by-step derivation

   8. Simplified 39.6%

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

Alternative 13: 38.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+108}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 0.0225:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.8e+108) t (if (<= z 0.0225) x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.8e+108) {
		tmp = t;
	} else if (z <= 0.0225) {
		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 <= (-2.8d+108)) then
        tmp = t
    else if (z <= 0.0225d0) 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 <= -2.8e+108) {
		tmp = t;
	} else if (z <= 0.0225) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.8e+108:
		tmp = t
	elif z <= 0.0225:
		tmp = x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.8e+108)
		tmp = t;
	elseif (z <= 0.0225)
		tmp = x;
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.8e+108)
		tmp = t;
	elseif (z <= 0.0225)
		tmp = x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.8e+108], t, If[LessEqual[z, 0.0225], x, t]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7999999999999998e108 or 0.022499999999999999 < z

   1. Initial program 64.2%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 54.4%

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

   if -2.7999999999999998e108 < z < 0.022499999999999999

   1. Initial program 90.1%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 40.6%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 25.0% 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 80.9%

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

3. Taylor expanded in z around inf

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

5. Simplified 25.1%

   \[\leadsto \color{blue}{t} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))