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

Percentage Accurate: 67.8% → 88.7%

Time: 13.0s

Alternatives: 18

Speedup: 0.8×

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: 67.8% 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: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(a - y\right) \cdot \frac{t - x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3e+91)
   (fma (- x t) (/ (- y a) z) t)
   (if (<= z 4e+202)
     (fma (- t x) (/ (- y z) (- a z)) x)
     (+ t (* (- a y) (/ (- t x) z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3e+91) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else if (z <= 4e+202) {
		tmp = fma((t - x), ((y - z) / (a - z)), x);
	} else {
		tmp = t + ((a - y) * ((t - x) / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3e+91)
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	elseif (z <= 4e+202)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / Float64(a - z)), x);
	else
		tmp = Float64(t + Float64(Float64(a - y) * Float64(Float64(t - x) / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3e+91], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 4e+202], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(t + N[(N[(a - y), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.00000000000000006e91

   1. Initial program 25.9%

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

   3. Taylor expanded in z around inf

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

      1. 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. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

   5. Simplified 80.3%

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

   if -3.00000000000000006e91 < z < 3.9999999999999996e202

   1. Initial program 84.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 94.9

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

   4. Applied egg-rr 94.9%

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

   if 3.9999999999999996e202 < z

   1. Initial program 18.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 56.6

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

   4. Applied egg-rr 56.6%

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

      1. div-sub N/A

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

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

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

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

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

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

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

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

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

      6. --lowering--.f64 56.6

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

   6. Applied egg-rr 56.6%

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

   7. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. --lowering--.f64 95.3

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

   9. Simplified 95.3%

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

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

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

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

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

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

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

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

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

       5. --lowering--.f64 95.4

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

   11. Applied egg-rr 95.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+202}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(a - y\right) \cdot \frac{t - x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 37.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{x}{a}, x\right)\\ \mathbf{if}\;a \leq -5.2 \cdot 10^{+69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -7.6 \cdot 10^{-135}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{-141}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{+145}:\\ \;\;\;\;t \cdot \frac{y - z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ x a) x)))
   (if (<= a -5.2e+69)
     t_1
     (if (<= a -7.6e-135)
       t
       (if (<= a 9.6e-141)
         (* x (/ y z))
         (if (<= a 1.85e+145) (* t (/ (- y z) a)) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (x / a), x);
	double tmp;
	if (a <= -5.2e+69) {
		tmp = t_1;
	} else if (a <= -7.6e-135) {
		tmp = t;
	} else if (a <= 9.6e-141) {
		tmp = x * (y / z);
	} else if (a <= 1.85e+145) {
		tmp = t * ((y - z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(x / a), x)
	tmp = 0.0
	if (a <= -5.2e+69)
		tmp = t_1;
	elseif (a <= -7.6e-135)
		tmp = t;
	elseif (a <= 9.6e-141)
		tmp = Float64(x * Float64(y / z));
	elseif (a <= 1.85e+145)
		tmp = Float64(t * Float64(Float64(y - z) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(x / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5.2e+69], t$95$1, If[LessEqual[a, -7.6e-135], t, If[LessEqual[a, 9.6e-141], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.85e+145], N[(t * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -5.2000000000000004e69 or 1.84999999999999997e145 < a

   1. Initial program 68.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 64.6

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

   5. Simplified 64.6%

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

   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 57.5%

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

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{x \cdot z}{a}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

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

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

       5. /-lowering-/.f64 56.6

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

   11. Simplified 56.6%

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

   if -5.2000000000000004e69 < a < -7.6000000000000005e-135

   1. Initial program 71.3%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 50.9%

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

   if -7.6000000000000005e-135 < a < 9.6000000000000004e-141

   1. Initial program 66.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 31.2

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

   5. Simplified 31.2%

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

   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. *-lowering-*.f64 N/A

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

       3. /-lowering-/.f64 47.1

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

   11. Simplified 47.1%

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

   if 9.6000000000000004e-141 < a < 1.84999999999999997e145

   1. Initial program 70.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 49.8

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

   5. Simplified 49.8%

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

   6. Taylor expanded in a around inf

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

      1. 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 36.1

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

   8. Simplified 36.1%

      \[\leadsto \color{blue}{t \cdot \frac{y - z}{a}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 64.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{a}{z}, t\right)\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-65}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t - t \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.5e+207)
   (fma (- t x) (/ a z) t)
   (if (<= z -1.85e-65)
     (* (- y z) (/ t (- a z)))
     (if (<= z 3.5e+42) (fma (- t x) (/ y a) x) (- t (* t (/ y z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.5e+207) {
		tmp = fma((t - x), (a / z), t);
	} else if (z <= -1.85e-65) {
		tmp = (y - z) * (t / (a - z));
	} else if (z <= 3.5e+42) {
		tmp = fma((t - x), (y / a), x);
	} else {
		tmp = t - (t * (y / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.5e+207)
		tmp = fma(Float64(t - x), Float64(a / z), t);
	elseif (z <= -1.85e-65)
		tmp = Float64(Float64(y - z) * Float64(t / Float64(a - z)));
	elseif (z <= 3.5e+42)
		tmp = fma(Float64(t - x), Float64(y / a), x);
	else
		tmp = Float64(t - Float64(t * Float64(y / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.5e+207], N[(N[(t - x), $MachinePrecision] * N[(a / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, -1.85e-65], N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.5e+42], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(t - N[(t * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8.4999999999999996e207

   1. Initial program 11.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 43.7

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

   4. Applied egg-rr 43.7%

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

      1. div-sub N/A

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

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

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

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

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

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

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

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

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

      6. --lowering--.f64 43.7

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

   6. Applied egg-rr 43.7%

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

   7. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. --lowering--.f64 79.8

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

   9. Simplified 79.8%

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

   10. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

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

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

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

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

       6. /-lowering-/.f64 58.0

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

   12. Simplified 58.0%

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

   if -8.4999999999999996e207 < z < -1.85e-65

   1. Initial program 70.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 50.6

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

   5. Simplified 50.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 58.9

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

   7. Applied egg-rr 58.9%

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

   if -1.85e-65 < z < 3.50000000000000023e42

   1. Initial program 90.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 96.7

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

   4. Applied egg-rr 96.7%

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

   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 74.9

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

   7. Simplified 74.9%

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

   if 3.50000000000000023e42 < z

   1. Initial program 43.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 43.4

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

   5. Simplified 43.4%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

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

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

      3. associate-/l* N/A

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

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

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

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

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

      6. --lowering--.f64 59.9

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

   8. Simplified 59.9%

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

   9. Taylor expanded in t around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. mul-1-neg N/A

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

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

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

       7. associate-/l* N/A

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

       8. unsub-neg N/A

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

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

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

       10. associate-/l* N/A

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

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

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

       12. /-lowering-/.f64 59.9

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

   11. Simplified 59.9%

       \[\leadsto \color{blue}{t - t \cdot \frac{y}{z}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -7 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-21}:\\ \;\;\;\;t + \left(a - y\right) \cdot \frac{t - x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- t x) a) x)))
   (if (<= a -7e+53)
     t_1
     (if (<= a 4.8e-21) (+ t (* (- a y) (/ (- t x) 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 <= -7e+53) {
		tmp = t_1;
	} else if (a <= 4.8e-21) {
		tmp = t + ((a - y) * ((t - x) / 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 <= -7e+53)
		tmp = t_1;
	elseif (a <= 4.8e-21)
		tmp = Float64(t + Float64(Float64(a - y) * Float64(Float64(t - x) / 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, -7e+53], t$95$1, If[LessEqual[a, 4.8e-21], N[(t + N[(N[(a - y), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.00000000000000038e53 or 4.7999999999999999e-21 < a

   1. Initial program 67.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 79.0

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

   5. Simplified 79.0%

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

   if -7.00000000000000038e53 < a < 4.7999999999999999e-21

   1. Initial program 70.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 79.6

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

   4. Applied egg-rr 79.6%

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

      1. div-sub N/A

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

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

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

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

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

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

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

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

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

      6. --lowering--.f64 79.6

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

   6. Applied egg-rr 79.6%

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

   7. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. --lowering--.f64 77.3

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

   9. Simplified 77.3%

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

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

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

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

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

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

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

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

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

       5. --lowering--.f64 77.3

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

   11. Applied egg-rr 77.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-21}:\\ \;\;\;\;t + \left(a - y\right) \cdot \frac{t - x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{+54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- t x) a) x)))
   (if (<= a -6.5e+54)
     t_1
     (if (<= a 4.7e-21) (fma (- x t) (/ (- y a) z) 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 <= -6.5e+54) {
		tmp = t_1;
	} else if (a <= 4.7e-21) {
		tmp = fma((x - t), ((y - a) / z), 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 <= -6.5e+54)
		tmp = t_1;
	elseif (a <= 4.7e-21)
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), 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, -6.5e+54], t$95$1, If[LessEqual[a, 4.7e-21], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.5e54 or 4.7000000000000003e-21 < a

   1. Initial program 67.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 79.0

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

   5. Simplified 79.0%

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

   if -6.5e54 < a < 4.7000000000000003e-21

   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 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. 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. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

   5. Simplified 80.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -2.6 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(a - y, \frac{t - x}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- t x) a) x)))
   (if (<= a -2.6e+44)
     t_1
     (if (<= a 4.8e-21) (fma (- a y) (/ (- t x) z) 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 <= -2.6e+44) {
		tmp = t_1;
	} else if (a <= 4.8e-21) {
		tmp = fma((a - y), ((t - x) / z), 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 <= -2.6e+44)
		tmp = t_1;
	elseif (a <= 4.8e-21)
		tmp = fma(Float64(a - y), Float64(Float64(t - x) / z), 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, -2.6e+44], t$95$1, If[LessEqual[a, 4.8e-21], N[(N[(a - y), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5999999999999999e44 or 4.7999999999999999e-21 < a

   1. Initial program 67.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 79.0

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

   5. Simplified 79.0%

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

   if -2.5999999999999999e44 < a < 4.7999999999999999e-21

   1. Initial program 70.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 79.6

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

   4. Applied egg-rr 79.6%

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

      1. div-sub N/A

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

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

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

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

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

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

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

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

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

      6. --lowering--.f64 79.6

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

   6. Applied egg-rr 79.6%

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

   7. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. --lowering--.f64 77.3

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

   9. Simplified 77.3%

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

Alternative 7: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a - y, \frac{t - x}{z}, t\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- a y) (/ (- t x) z) t)))
   (if (<= z -4e-64) t_1 (if (<= z 1.02e+15) (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((a - y), ((t - x) / z), t);
	double tmp;
	if (z <= -4e-64) {
		tmp = t_1;
	} else if (z <= 1.02e+15) {
		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(a - y), Float64(Float64(t - x) / z), t)
	tmp = 0.0
	if (z <= -4e-64)
		tmp = t_1;
	elseif (z <= 1.02e+15)
		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[(a - y), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -4e-64], t$95$1, If[LessEqual[z, 1.02e+15], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.99999999999999986e-64 or 1.02e15 < z

   1. Initial program 50.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 75.5

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

   4. Applied egg-rr 75.5%

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

      1. div-sub N/A

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

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

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

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

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

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

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

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

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

      6. --lowering--.f64 75.5

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

   6. Applied egg-rr 75.5%

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

   7. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. --lowering--.f64 73.7

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

   9. Simplified 73.7%

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

   if -3.99999999999999986e-64 < z < 1.02e15

   1. Initial program 90.4%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 96.6

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

   4. Applied egg-rr 96.6%

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

   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 75.5

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

   7. Simplified 75.5%

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

Alternative 8: 38.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{a}, x\right)\\ \mathbf{elif}\;a \leq -2.15 \cdot 10^{-134}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 2.85 \cdot 10^{-168}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 380000000000:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.9e+74)
   (fma z (/ x a) x)
   (if (<= a -2.15e-134)
     t
     (if (<= a 2.85e-168) (* x (/ y z)) (if (<= a 380000000000.0) t x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.9e+74) {
		tmp = fma(z, (x / a), x);
	} else if (a <= -2.15e-134) {
		tmp = t;
	} else if (a <= 2.85e-168) {
		tmp = x * (y / z);
	} else if (a <= 380000000000.0) {
		tmp = t;
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.9e+74)
		tmp = fma(z, Float64(x / a), x);
	elseif (a <= -2.15e-134)
		tmp = t;
	elseif (a <= 2.85e-168)
		tmp = Float64(x * Float64(y / z));
	elseif (a <= 380000000000.0)
		tmp = t;
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.9e+74], N[(z * N[(x / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, -2.15e-134], t, If[LessEqual[a, 2.85e-168], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 380000000000.0], t, x]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.8999999999999999e74

   1. Initial program 64.2%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 60.2

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

   5. Simplified 60.2%

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

   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 49.0%

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

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{x \cdot z}{a}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

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

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

       5. /-lowering-/.f64 47.6

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

   11. Simplified 47.6%

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

   if -1.8999999999999999e74 < a < -2.14999999999999993e-134 or 2.85000000000000004e-168 < a < 3.8e11

   1. Initial program 73.6%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 42.5%

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

   if -2.14999999999999993e-134 < a < 2.85000000000000004e-168

   1. Initial program 68.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 32.5

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

   5. Simplified 32.5%

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

   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 31.9

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

   8. Simplified 31.9%

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

   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. *-lowering-*.f64 N/A

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

       3. /-lowering-/.f64 49.0

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

   11. Simplified 49.0%

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

   if 3.8e11 < a

   1. Initial program 68.6%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 43.4%

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

Alternative 9: 38.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+69}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -7.6 \cdot 10^{-135}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-171}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 170000000000:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4.8e+69)
   x
   (if (<= a -7.6e-135)
     t
     (if (<= a 1.85e-171) (* x (/ y z)) (if (<= a 170000000000.0) t x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.8e+69) {
		tmp = x;
	} else if (a <= -7.6e-135) {
		tmp = t;
	} else if (a <= 1.85e-171) {
		tmp = x * (y / z);
	} else if (a <= 170000000000.0) {
		tmp = t;
	} else {
		tmp = x;
	}
	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 (a <= (-4.8d+69)) then
        tmp = x
    else if (a <= (-7.6d-135)) then
        tmp = t
    else if (a <= 1.85d-171) then
        tmp = x * (y / z)
    else if (a <= 170000000000.0d0) then
        tmp = t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.8e+69) {
		tmp = x;
	} else if (a <= -7.6e-135) {
		tmp = t;
	} else if (a <= 1.85e-171) {
		tmp = x * (y / z);
	} else if (a <= 170000000000.0) {
		tmp = t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -4.8e+69:
		tmp = x
	elif a <= -7.6e-135:
		tmp = t
	elif a <= 1.85e-171:
		tmp = x * (y / z)
	elif a <= 170000000000.0:
		tmp = t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4.8e+69)
		tmp = x;
	elseif (a <= -7.6e-135)
		tmp = t;
	elseif (a <= 1.85e-171)
		tmp = Float64(x * Float64(y / z));
	elseif (a <= 170000000000.0)
		tmp = t;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -4.8e+69)
		tmp = x;
	elseif (a <= -7.6e-135)
		tmp = t;
	elseif (a <= 1.85e-171)
		tmp = x * (y / z);
	elseif (a <= 170000000000.0)
		tmp = t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4.8e+69], x, If[LessEqual[a, -7.6e-135], t, If[LessEqual[a, 1.85e-171], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 170000000000.0], t, x]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.8000000000000003e69 or 1.7e11 < a

   1. Initial program 66.7%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 45.2%

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

   if -4.8000000000000003e69 < a < -7.6000000000000005e-135 or 1.85000000000000006e-171 < a < 1.7e11

   1. Initial program 73.6%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 42.5%

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

   if -7.6000000000000005e-135 < a < 1.85000000000000006e-171

   1. Initial program 68.0%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 32.5

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

   5. Simplified 32.5%

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

   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 31.9

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

   8. Simplified 31.9%

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

   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. *-lowering-*.f64 N/A

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

       3. /-lowering-/.f64 49.0

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

   11. Simplified 49.0%

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

Alternative 10: 70.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a - y, -\frac{x}{z}, t\right)\\ \mathbf{if}\;z \leq -0.72:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- a y) (- (/ x z)) t)))
   (if (<= z -0.72) t_1 (if (<= z 1.1e+50) (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((a - y), -(x / z), t);
	double tmp;
	if (z <= -0.72) {
		tmp = t_1;
	} else if (z <= 1.1e+50) {
		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(a - y), Float64(-Float64(x / z)), t)
	tmp = 0.0
	if (z <= -0.72)
		tmp = t_1;
	elseif (z <= 1.1e+50)
		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[(a - y), $MachinePrecision] * (-N[(x / z), $MachinePrecision]) + t), $MachinePrecision]}, If[LessEqual[z, -0.72], t$95$1, If[LessEqual[z, 1.1e+50], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.71999999999999997 or 1.10000000000000008e50 < z

   1. Initial program 43.3%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 71.1

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

   4. Applied egg-rr 71.1%

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

      1. div-sub N/A

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

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

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

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

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

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

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

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

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

      6. --lowering--.f64 71.1

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

   6. Applied egg-rr 71.1%

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

   7. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. --lowering--.f64 76.7

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

   9. Simplified 76.7%

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

   10. Taylor expanded in t around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 71.7

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

   12. Simplified 71.7%

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

   if -0.71999999999999997 < z < 1.10000000000000008e50

   1. Initial program 90.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 97.1

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

   4. Applied egg-rr 97.1%

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

   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 72.3

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

   7. Simplified 72.3%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.72:\\ \;\;\;\;\mathsf{fma}\left(a - y, -\frac{x}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - y, -\frac{x}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{a}{z}, t\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t - t \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.8e+71)
   (fma (- t x) (/ a z) t)
   (if (<= z 7.2e+47) (fma (- t x) (/ y a) x) (- t (* t (/ y z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.8e+71) {
		tmp = fma((t - x), (a / z), t);
	} else if (z <= 7.2e+47) {
		tmp = fma((t - x), (y / a), x);
	} else {
		tmp = t - (t * (y / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.8e+71)
		tmp = fma(Float64(t - x), Float64(a / z), t);
	elseif (z <= 7.2e+47)
		tmp = fma(Float64(t - x), Float64(y / a), x);
	else
		tmp = Float64(t - Float64(t * Float64(y / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.8e+71], N[(N[(t - x), $MachinePrecision] * N[(a / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 7.2e+47], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(t - N[(t * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.8000000000000002e71

   1. Initial program 30.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 60.0

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

   4. Applied egg-rr 60.0%

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

      1. div-sub N/A

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

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

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

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

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

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

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

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

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

      6. --lowering--.f64 60.0

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

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. --lowering--.f64 77.7

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

   9. Simplified 77.7%

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

   10. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

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

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

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

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

       6. /-lowering-/.f64 54.6

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

   12. Simplified 54.6%

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

   if -7.8000000000000002e71 < z < 7.20000000000000015e47

   1. Initial program 89.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 96.1

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

   4. Applied egg-rr 96.1%

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

   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 68.0

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

   7. Simplified 68.0%

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

   if 7.20000000000000015e47 < z

   1. Initial program 43.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 43.4

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

   5. Simplified 43.4%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

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

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

      3. associate-/l* N/A

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

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

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

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

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

      6. --lowering--.f64 59.9

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

   8. Simplified 59.9%

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

   9. Taylor expanded in t around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. mul-1-neg N/A

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

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

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

       7. associate-/l* N/A

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

       8. unsub-neg N/A

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

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

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

       10. associate-/l* N/A

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

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

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

       12. /-lowering-/.f64 59.9

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

   11. Simplified 59.9%

       \[\leadsto \color{blue}{t - t \cdot \frac{y}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 63.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{a}{z}, t\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t - t \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.6e+68)
   (fma (- t x) (/ a z) t)
   (if (<= z 2.3e+39) (fma y (/ (- t x) a) x) (- t (* t (/ y z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.6e+68) {
		tmp = fma((t - x), (a / z), t);
	} else if (z <= 2.3e+39) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t - (t * (y / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.6e+68)
		tmp = fma(Float64(t - x), Float64(a / z), t);
	elseif (z <= 2.3e+39)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = Float64(t - Float64(t * Float64(y / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.6e+68], N[(N[(t - x), $MachinePrecision] * N[(a / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 2.3e+39], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(t - N[(t * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.6000000000000002e68

   1. Initial program 30.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 60.0

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

   4. Applied egg-rr 60.0%

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

      1. div-sub N/A

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

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

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

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

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

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

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

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

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

      6. --lowering--.f64 60.0

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

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. --lowering--.f64 77.7

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

   9. Simplified 77.7%

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

   10. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

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

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

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

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

       6. /-lowering-/.f64 54.6

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

   12. Simplified 54.6%

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

   if -7.6000000000000002e68 < z < 2.30000000000000012e39

   1. Initial program 89.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 65.5

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

   5. Simplified 65.5%

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

   if 2.30000000000000012e39 < z

   1. Initial program 43.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 43.4

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

   5. Simplified 43.4%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

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

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

      3. associate-/l* N/A

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

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

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

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

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

      6. --lowering--.f64 59.9

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

   8. Simplified 59.9%

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

   9. Taylor expanded in t around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. mul-1-neg N/A

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

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

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

       7. associate-/l* N/A

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

       8. unsub-neg N/A

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

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

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

       10. associate-/l* N/A

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

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

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

       12. /-lowering-/.f64 59.9

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

   11. Simplified 59.9%

       \[\leadsto \color{blue}{t - t \cdot \frac{y}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 63.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t - t \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.8e+64)
   (fma a (/ (- t x) z) t)
   (if (<= z 1.5e+41) (fma y (/ (- t x) a) x) (- t (* t (/ y z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.8e+64) {
		tmp = fma(a, ((t - x) / z), t);
	} else if (z <= 1.5e+41) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t - (t * (y / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.8e+64)
		tmp = fma(a, Float64(Float64(t - x) / z), t);
	elseif (z <= 1.5e+41)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = Float64(t - Float64(t * Float64(y / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.8e+64], N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 1.5e+41], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(t - N[(t * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.7999999999999996e64

   1. Initial program 30.5%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 60.0

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

   4. Applied egg-rr 60.0%

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

      1. div-sub N/A

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

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

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

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

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

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

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

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

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

      6. --lowering--.f64 60.0

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

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. --lowering--.f64 77.7

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

   9. Simplified 77.7%

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

   10. Taylor expanded in a around inf

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

   12. Simplified 52.9%

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

   if -7.7999999999999996e64 < z < 1.4999999999999999e41

   1. Initial program 89.1%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 65.5

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

   5. Simplified 65.5%

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

   if 1.4999999999999999e41 < z

   1. Initial program 43.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 43.4

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

   5. Simplified 43.4%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

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

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

      3. associate-/l* N/A

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

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

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

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

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

      6. --lowering--.f64 59.9

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

   8. Simplified 59.9%

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

   9. Taylor expanded in t around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. mul-1-neg N/A

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

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

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

       7. associate-/l* N/A

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

       8. unsub-neg N/A

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

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

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

       10. associate-/l* N/A

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

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

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

       12. /-lowering-/.f64 59.9

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

   11. Simplified 59.9%

       \[\leadsto \color{blue}{t - t \cdot \frac{y}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 50.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{a}, x\right)\\ \mathbf{elif}\;a \leq 290000000000:\\ \;\;\;\;t - t \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -3.3e+69)
   (fma z (/ x a) x)
   (if (<= a 290000000000.0) (- t (* t (/ y z))) (- x (/ (* x y) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -3.3e+69) {
		tmp = fma(z, (x / a), x);
	} else if (a <= 290000000000.0) {
		tmp = t - (t * (y / z));
	} else {
		tmp = x - ((x * y) / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -3.3e+69)
		tmp = fma(z, Float64(x / a), x);
	elseif (a <= 290000000000.0)
		tmp = Float64(t - Float64(t * Float64(y / z)));
	else
		tmp = Float64(x - Float64(Float64(x * y) / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -3.3e+69], N[(z * N[(x / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 290000000000.0], N[(t - N[(t * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.2999999999999999e69

   1. Initial program 64.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 59.0

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

   5. Simplified 59.0%

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

   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 48.0%

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

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{x \cdot z}{a}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

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

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

       5. /-lowering-/.f64 46.7

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

   11. Simplified 46.7%

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

   if -3.2999999999999999e69 < a < 2.9e11

   1. Initial program 70.8%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 54.9

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

   5. Simplified 54.9%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

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

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

      3. associate-/l* N/A

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

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

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

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

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

      6. --lowering--.f64 54.3

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

   8. Simplified 54.3%

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

   9. Taylor expanded in t around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. mul-1-neg N/A

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

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

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

       7. associate-/l* N/A

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

       8. unsub-neg N/A

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

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

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

       10. associate-/l* N/A

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

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

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

       12. /-lowering-/.f64 54.3

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

   11. Simplified 54.3%

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

   if 2.9e11 < a

   1. Initial program 68.6%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 52.6

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

   5. Simplified 52.6%

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

   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 45.2

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

   8. Simplified 45.2%

      \[\leadsto \color{blue}{x - \frac{x \cdot y}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 50.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{a}, x\right)\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+132}:\\ \;\;\;\;t - t \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -3e+67)
   (fma z (/ x a) x)
   (if (<= a 9.6e+132) (- t (* t (/ y z))) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -3e+67) {
		tmp = fma(z, (x / a), x);
	} else if (a <= 9.6e+132) {
		tmp = t - (t * (y / z));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -3e+67)
		tmp = fma(z, Float64(x / a), x);
	elseif (a <= 9.6e+132)
		tmp = Float64(t - Float64(t * Float64(y / z)));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -3e+67], N[(z * N[(x / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 9.6e+132], N[(t - N[(t * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if a < -3.0000000000000001e67

   1. Initial program 64.9%

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
   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 59.0

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

   5. Simplified 59.0%

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

   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 48.0%

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

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{x \cdot z}{a}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

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

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

       5. /-lowering-/.f64 46.7

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

   11. Simplified 46.7%

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

   if -3.0000000000000001e67 < a < 9.6000000000000004e132

   1. Initial program 69.9%

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

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

      1. mul-1-neg N/A

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

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

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

      3. associate-/l* N/A

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

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

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

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

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

      6. --lowering--.f64 48.5

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

   8. Simplified 48.5%

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

   9. Taylor expanded in t around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. mul-1-neg N/A

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

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

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

       7. associate-/l* N/A

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

       8. unsub-neg N/A

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

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

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

       10. associate-/l* N/A

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

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

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

       12. /-lowering-/.f64 48.5

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

   11. Simplified 48.5%

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

   if 9.6000000000000004e132 < a

   1. Initial program 71.3%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 67.9%

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

Alternative 16: 39.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 4000000000000:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.15e+72) x (if (<= a 4000000000000.0) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.15e+72) {
		tmp = x;
	} else if (a <= 4000000000000.0) {
		tmp = t;
	} else {
		tmp = x;
	}
	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 (a <= (-1.15d+72)) then
        tmp = x
    else if (a <= 4000000000000.0d0) then
        tmp = t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.15e+72) {
		tmp = x;
	} else if (a <= 4000000000000.0) {
		tmp = t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.15e+72:
		tmp = x
	elif a <= 4000000000000.0:
		tmp = t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.15e+72)
		tmp = x;
	elseif (a <= 4000000000000.0)
		tmp = t;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.15e+72)
		tmp = x;
	elseif (a <= 4000000000000.0)
		tmp = t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.15e+72], x, If[LessEqual[a, 4000000000000.0], t, x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.15e72 or 4e12 < a

   1. Initial program 66.7%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 45.2%

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

   if -1.15e72 < a < 4e12

   1. Initial program 71.0%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 36.9%

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

Alternative 17: 25.4% 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 69.1%

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

3. Taylor expanded in z around inf

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

5. Simplified 24.4%

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

Alternative 18: 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 69.1%

   \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
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 41.9

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

5. Simplified 41.9%

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

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: 83.9% 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 2024204 
(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))))