Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A

Percentage Accurate: 87.8% → 99.8%

Time: 8.7s

Alternatives: 10

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))

C

double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + y) / (1.0d0 - (y / z))
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Python

def code(x, y, z):
	return (x + y) / (1.0 - (y / z))

Julia

function code(x, y, z)
	return Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) / (1.0 - (y / z));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))

C

double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + y) / (1.0d0 - (y / z))
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Python

def code(x, y, z):
	return (x + y) / (1.0 - (y / z))

Julia

function code(x, y, z)
	return Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) / (1.0 - (y / z));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-282}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(x + y\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -4e-282)
     (* (/ z (- z y)) (+ x y))
     (if (<= t_0 0.0) (- (fma z (/ x y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -4e-282) {
		tmp = (z / (z - y)) * (x + y);
	} else if (t_0 <= 0.0) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -4e-282)
		tmp = Float64(Float64(z / Float64(z - y)) * Float64(x + y));
	elseif (t_0 <= 0.0)
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-282], N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -4.0000000000000001e-282

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}} \]
   4. Step-by-step derivation

      1. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. *-inverses N/A

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

      8. div-sub N/A

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

      9. associate-/r/ N/A

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

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

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

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

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

      12. --lowering--.f64 93.9

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

   5. Simplified 93.9%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-out N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 99.8

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

   8. Simplified 99.8%

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

   if -4.0000000000000001e-282 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 5.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. distribute-neg-frac N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. neg-mul-1 N/A

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

      8. unsub-neg N/A

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

      9. div-sub N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

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

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

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

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-commutative N/A

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

      20. unsub-neg N/A

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

      21. mul-1-neg N/A

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

      22. distribute-neg-out N/A

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

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

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

   5. Simplified 99.9%

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

   if 0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -4 \cdot 10^{-282}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(x + y\right)\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ t_1 := \frac{z}{z - y} \cdot \left(x + y\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-282}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))) (t_1 (* (/ z (- z y)) (+ x y))))
   (if (<= t_0 -4e-282) t_1 (if (<= t_0 0.0) (- (fma z (/ x y) z)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double t_1 = (z / (z - y)) * (x + y);
	double tmp;
	if (t_0 <= -4e-282) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	t_1 = Float64(Float64(z / Float64(z - y)) * Float64(x + y))
	tmp = 0.0
	if (t_0 <= -4e-282)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-282], t$95$1, If[LessEqual[t$95$0, 0.0], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -4.0000000000000001e-282 or 0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}} \]
   4. Step-by-step derivation

      1. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

      7. *-inverses N/A

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

      8. div-sub N/A

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

      9. associate-/r/ N/A

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

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

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

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

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

      12. --lowering--.f64 93.1

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

   5. Simplified 93.1%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-out N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 99.7

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

   8. Simplified 99.7%

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

   if -4.0000000000000001e-282 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 5.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. distribute-neg-frac N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. neg-mul-1 N/A

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

      8. unsub-neg N/A

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

      9. div-sub N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

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

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

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

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-commutative N/A

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

      20. unsub-neg N/A

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

      21. mul-1-neg N/A

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

      22. distribute-neg-out N/A

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

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

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

   5. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -4 \cdot 10^{-282}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(x + y\right)\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+35}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -2.25 \cdot 10^{-218}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-181}:\\ \;\;\;\;\frac{x \cdot z}{-y}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-84}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.75e+35)
   (+ x y)
   (if (<= z -2.25e-218)
     (- z)
     (if (<= z 8.5e-181)
       (/ (* x z) (- y))
       (if (<= z 4.6e-84) (- z) (+ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.75e+35) {
		tmp = x + y;
	} else if (z <= -2.25e-218) {
		tmp = -z;
	} else if (z <= 8.5e-181) {
		tmp = (x * z) / -y;
	} else if (z <= 4.6e-84) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.75d+35)) then
        tmp = x + y
    else if (z <= (-2.25d-218)) then
        tmp = -z
    else if (z <= 8.5d-181) then
        tmp = (x * z) / -y
    else if (z <= 4.6d-84) then
        tmp = -z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.75e+35) {
		tmp = x + y;
	} else if (z <= -2.25e-218) {
		tmp = -z;
	} else if (z <= 8.5e-181) {
		tmp = (x * z) / -y;
	} else if (z <= 4.6e-84) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.75e+35:
		tmp = x + y
	elif z <= -2.25e-218:
		tmp = -z
	elif z <= 8.5e-181:
		tmp = (x * z) / -y
	elif z <= 4.6e-84:
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.75e+35)
		tmp = Float64(x + y);
	elseif (z <= -2.25e-218)
		tmp = Float64(-z);
	elseif (z <= 8.5e-181)
		tmp = Float64(Float64(x * z) / Float64(-y));
	elseif (z <= 4.6e-84)
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.75e+35)
		tmp = x + y;
	elseif (z <= -2.25e-218)
		tmp = -z;
	elseif (z <= 8.5e-181)
		tmp = (x * z) / -y;
	elseif (z <= 4.6e-84)
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.75e+35], N[(x + y), $MachinePrecision], If[LessEqual[z, -2.25e-218], (-z), If[LessEqual[z, 8.5e-181], N[(N[(x * z), $MachinePrecision] / (-y)), $MachinePrecision], If[LessEqual[z, 4.6e-84], (-z), N[(x + y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.75e35 or 4.59999999999999961e-84 < z

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 72.8

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

   5. Simplified 72.8%

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

   if -1.75e35 < z < -2.24999999999999988e-218 or 8.49999999999999953e-181 < z < 4.59999999999999961e-84

   1. Initial program 81.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 59.0

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 59.0%

      \[\leadsto \color{blue}{-z} \]

   if -2.24999999999999988e-218 < z < 8.49999999999999953e-181

   1. Initial program 62.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 - \frac{y}{z}}} \]
   4. Step-by-step derivation

      1. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

      6. --lowering--.f64 51.9

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

   5. Simplified 51.9%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 55.3

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

   8. Simplified 55.3%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+35}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -2.25 \cdot 10^{-218}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-181}:\\ \;\;\;\;\frac{x \cdot z}{-y}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-84}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \mathsf{fma}\left(\frac{y}{z}, y, x\right)\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+65}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (fma (/ y z) y x))))
   (if (<= z -1.95e+103)
     t_0
     (if (<= z 2.25e+65) (- (- z) (/ (* x z) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y + fma((y / z), y, x);
	double tmp;
	if (z <= -1.95e+103) {
		tmp = t_0;
	} else if (z <= 2.25e+65) {
		tmp = -z - ((x * z) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + fma(Float64(y / z), y, x))
	tmp = 0.0
	if (z <= -1.95e+103)
		tmp = t_0;
	elseif (z <= 2.25e+65)
		tmp = Float64(Float64(-z) - Float64(Float64(x * z) / y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(N[(y / z), $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.95e+103], t$95$0, If[LessEqual[z, 2.25e+65], N[((-z) - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9499999999999999e103 or 2.25e65 < z

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 87.4

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

   5. Simplified 87.4%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 87.5%

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

   if -1.9499999999999999e103 < z < 2.25e65

   1. Initial program 80.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) - \frac{{z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) + \left(\mathsf{neg}\left(\frac{{z}^{2}}{y}\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot z}{y}\right)\right)}\right) + \left(\mathsf{neg}\left(\frac{{z}^{2}}{y}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot z - \frac{x \cdot z}{y}\right)} + \left(\mathsf{neg}\left(\frac{{z}^{2}}{y}\right)\right) \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{-1 \cdot z - \left(\frac{x \cdot z}{y} - \left(\mathsf{neg}\left(\frac{{z}^{2}}{y}\right)\right)\right)} \]

      5. distribute-frac-neg N/A

         \[\leadsto -1 \cdot z - \left(\frac{x \cdot z}{y} - \color{blue}{\frac{\mathsf{neg}\left({z}^{2}\right)}{y}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto -1 \cdot z - \left(\frac{x \cdot z}{y} - \frac{\color{blue}{-1 \cdot {z}^{2}}}{y}\right) \]

      7. div-sub N/A

         \[\leadsto -1 \cdot z - \color{blue}{\frac{x \cdot z - -1 \cdot {z}^{2}}{y}} \]

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

         \[\leadsto \color{blue}{-1 \cdot z - \frac{x \cdot z - -1 \cdot {z}^{2}}{y}} \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - \frac{x \cdot z - -1 \cdot {z}^{2}}{y} \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - \frac{x \cdot z - -1 \cdot {z}^{2}}{y} \]

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

          \[\leadsto \left(\mathsf{neg}\left(z\right)\right) - \color{blue}{\frac{x \cdot z - -1 \cdot {z}^{2}}{y}} \]

      12. cancel-sign-sub-inv N/A

          \[\leadsto \left(\mathsf{neg}\left(z\right)\right) - \frac{\color{blue}{x \cdot z + \left(\mathsf{neg}\left(-1\right)\right) \cdot {z}^{2}}}{y} \]

      13. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(z\right)\right) - \frac{x \cdot z + \color{blue}{1} \cdot {z}^{2}}{y} \]

      14. *-lft-identity N/A

          \[\leadsto \left(\mathsf{neg}\left(z\right)\right) - \frac{x \cdot z + \color{blue}{{z}^{2}}}{y} \]

      15. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(z\right)\right) - \frac{\color{blue}{{z}^{2} + x \cdot z}}{y} \]

      16. unpow2 N/A

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

      17. distribute-rgt-out N/A

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

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

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

      19. +-lowering-+.f64 73.0

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

   5. Simplified 73.0%

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

   6. Taylor expanded in z around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 73.1

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

   8. Simplified 73.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+103}:\\ \;\;\;\;y + \mathsf{fma}\left(\frac{y}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+65}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(\frac{y}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \mathsf{fma}\left(\frac{y}{z}, y, x\right)\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+67}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (fma (/ y z) y x))))
   (if (<= z -1.95e+103) t_0 (if (<= z 5.1e+67) (- (fma z (/ x y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y + fma((y / z), y, x);
	double tmp;
	if (z <= -1.95e+103) {
		tmp = t_0;
	} else if (z <= 5.1e+67) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + fma(Float64(y / z), y, x))
	tmp = 0.0
	if (z <= -1.95e+103)
		tmp = t_0;
	elseif (z <= 5.1e+67)
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(N[(y / z), $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.95e+103], t$95$0, If[LessEqual[z, 5.1e+67], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9499999999999999e103 or 5.1000000000000002e67 < z

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 87.4

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

   5. Simplified 87.4%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 87.5%

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

   if -1.9499999999999999e103 < z < 5.1000000000000002e67

   1. Initial program 80.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. distribute-neg-frac N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. neg-mul-1 N/A

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

      8. unsub-neg N/A

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

      9. div-sub N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

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

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

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

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-commutative N/A

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

      20. unsub-neg N/A

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

      21. mul-1-neg N/A

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

      22. distribute-neg-out N/A

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

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

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

   5. Simplified 72.4%

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

Alternative 6: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.95 \cdot 10^{+91}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+65}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.95e+91)
   (+ x y)
   (if (<= z 5.8e+65) (- (fma z (/ x y) z)) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.95e+91) {
		tmp = x + y;
	} else if (z <= 5.8e+65) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.95e+91)
		tmp = Float64(x + y);
	elseif (z <= 5.8e+65)
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.95e+91], N[(x + y), $MachinePrecision], If[LessEqual[z, 5.8e+65], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9500000000000001e91 or 5.8000000000000001e65 < z

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 85.2

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

   5. Simplified 85.2%

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

   if -2.9500000000000001e91 < z < 5.8000000000000001e65

   1. Initial program 80.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. distribute-neg-frac N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. neg-mul-1 N/A

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

      8. unsub-neg N/A

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

      9. div-sub N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

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

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

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

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-commutative N/A

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

      20. unsub-neg N/A

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

      21. mul-1-neg N/A

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

      22. distribute-neg-out N/A

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

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

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

   5. Simplified 72.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.95 \cdot 10^{+91}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+65}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4200000:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-26}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.14 \cdot 10^{+114}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4200000.0)
   (- z)
   (if (<= y 3.7e-26) x (if (<= y 1.14e+114) y (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4200000.0) {
		tmp = -z;
	} else if (y <= 3.7e-26) {
		tmp = x;
	} else if (y <= 1.14e+114) {
		tmp = y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4200000.0d0)) then
        tmp = -z
    else if (y <= 3.7d-26) then
        tmp = x
    else if (y <= 1.14d+114) then
        tmp = y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4200000.0) {
		tmp = -z;
	} else if (y <= 3.7e-26) {
		tmp = x;
	} else if (y <= 1.14e+114) {
		tmp = y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4200000.0:
		tmp = -z
	elif y <= 3.7e-26:
		tmp = x
	elif y <= 1.14e+114:
		tmp = y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4200000.0)
		tmp = Float64(-z);
	elseif (y <= 3.7e-26)
		tmp = x;
	elseif (y <= 1.14e+114)
		tmp = y;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4200000.0)
		tmp = -z;
	elseif (y <= 3.7e-26)
		tmp = x;
	elseif (y <= 1.14e+114)
		tmp = y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4200000.0], (-z), If[LessEqual[y, 3.7e-26], x, If[LessEqual[y, 1.14e+114], y, (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.2e6 or 1.14e114 < y

   1. Initial program 76.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 54.0

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 54.0%

      \[\leadsto \color{blue}{-z} \]

   if -4.2e6 < y < 3.6999999999999999e-26

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Simplified 56.1%

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

   if 3.6999999999999999e-26 < y < 1.14e114

   1. Initial program 93.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 42.9

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

   5. Simplified 42.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y} \]
   7. Step-by-step derivation

   8. Simplified 39.9%

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

Alternative 8: 59.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+33}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-78}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.7e+33) (+ x y) (if (<= z 2.4e-78) (- z) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.7e+33) {
		tmp = x + y;
	} else if (z <= 2.4e-78) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.7d+33)) then
        tmp = x + y
    else if (z <= 2.4d-78) then
        tmp = -z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.7e+33) {
		tmp = x + y;
	} else if (z <= 2.4e-78) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.7e+33:
		tmp = x + y
	elif z <= 2.4e-78:
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.7e+33)
		tmp = Float64(x + y);
	elseif (z <= 2.4e-78)
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.7e+33)
		tmp = x + y;
	elseif (z <= 2.4e-78)
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.7e+33], N[(x + y), $MachinePrecision], If[LessEqual[z, 2.4e-78], (-z), N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.69999999999999991e33 or 2.4e-78 < z

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 72.8

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

   5. Simplified 72.8%

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

   if -2.69999999999999991e33 < z < 2.4e-78

   1. Initial program 73.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 49.0

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 49.0%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+33}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-78}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-129}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-125}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.3e-129) x (if (<= x 1.5e-125) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.3e-129) {
		tmp = x;
	} else if (x <= 1.5e-125) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-5.3d-129)) then
        tmp = x
    else if (x <= 1.5d-125) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.3e-129) {
		tmp = x;
	} else if (x <= 1.5e-125) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.3e-129:
		tmp = x
	elif x <= 1.5e-125:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.3e-129)
		tmp = x;
	elseif (x <= 1.5e-125)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.3e-129)
		tmp = x;
	elseif (x <= 1.5e-125)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.3e-129], x, If[LessEqual[x, 1.5e-125], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.29999999999999974e-129 or 1.49999999999999995e-125 < x

   1. Initial program 89.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Simplified 36.2%

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

   if -5.29999999999999974e-129 < x < 1.49999999999999995e-125

   1. Initial program 84.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 54.8

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

   5. Simplified 54.8%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y} \]
   7. Step-by-step derivation

   8. Simplified 48.3%

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

Alternative 10: 34.1% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 88.1%

   \[\frac{x + y}{1 - \frac{y}{z}} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

5. Simplified 28.1%

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

Developer Target 1: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{-y} \cdot z\\ \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (+ y x) (- y)) z)))
   (if (< y -3.7429310762689856e+171)
     t_0
     (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y + x) / -y) * z
    if (y < (-3.7429310762689856d+171)) then
        tmp = t_0
    else if (y < 3.5534662456086734d+168) then
        tmp = (x + y) / (1.0d0 - (y / z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y + x) / -y) * z
	tmp = 0
	if y < -3.7429310762689856e+171:
		tmp = t_0
	elif y < 3.5534662456086734e+168:
		tmp = (x + y) / (1.0 - (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y + x) / Float64(-y)) * z)
	tmp = 0.0
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y + x) / -y) * z;
	tmp = 0.0;
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = (x + y) / (1.0 - (y / z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] / (-y)), $MachinePrecision] * z), $MachinePrecision]}, If[Less[y, -3.7429310762689856e+171], t$95$0, If[Less[y, 3.5534662456086734e+168], N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -3742931076268985600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (/ (+ y x) (- y)) z) (if (< y 3553466245608673400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z))))

  (/ (+ x y) (- 1.0 (/ y z))))