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

Percentage Accurate: 87.8% → 99.8%

Time: 8.9s

Alternatives: 8

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. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.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. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.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. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.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. lower-+.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. lower-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. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.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. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.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. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.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. lower-+.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. lower-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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+33}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-215}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{x \cdot z}{-y}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-76}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8e+33)
   (+ x y)
   (if (<= z -1.45e-215)
     (- z)
     (if (<= z 7.2e-190)
       (/ (* x z) (- y))
       (if (<= z 3.4e-76) (- z) (+ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8e+33) {
		tmp = x + y;
	} else if (z <= -1.45e-215) {
		tmp = -z;
	} else if (z <= 7.2e-190) {
		tmp = (x * z) / -y;
	} else if (z <= 3.4e-76) {
		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 <= (-8d+33)) then
        tmp = x + y
    else if (z <= (-1.45d-215)) then
        tmp = -z
    else if (z <= 7.2d-190) then
        tmp = (x * z) / -y
    else if (z <= 3.4d-76) 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 <= -8e+33) {
		tmp = x + y;
	} else if (z <= -1.45e-215) {
		tmp = -z;
	} else if (z <= 7.2e-190) {
		tmp = (x * z) / -y;
	} else if (z <= 3.4e-76) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8e+33:
		tmp = x + y
	elif z <= -1.45e-215:
		tmp = -z
	elif z <= 7.2e-190:
		tmp = (x * z) / -y
	elif z <= 3.4e-76:
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8e+33)
		tmp = Float64(x + y);
	elseif (z <= -1.45e-215)
		tmp = Float64(-z);
	elseif (z <= 7.2e-190)
		tmp = Float64(Float64(x * z) / Float64(-y));
	elseif (z <= 3.4e-76)
		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 <= -8e+33)
		tmp = x + y;
	elseif (z <= -1.45e-215)
		tmp = -z;
	elseif (z <= 7.2e-190)
		tmp = (x * z) / -y;
	elseif (z <= 3.4e-76)
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8e+33], N[(x + y), $MachinePrecision], If[LessEqual[z, -1.45e-215], (-z), If[LessEqual[z, 7.2e-190], N[(N[(x * z), $MachinePrecision] / (-y)), $MachinePrecision], If[LessEqual[z, 3.4e-76], (-z), N[(x + y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.9999999999999996e33 or 3.3999999999999999e-76 < 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. lower-+.f64 72.8

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

   5. Simplified 72.8%

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

   if -7.9999999999999996e33 < z < -1.45e-215 or 7.20000000000000014e-190 < z < 3.3999999999999999e-76

   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. lower-neg.f64 59.0

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

   5. Simplified 59.0%

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

   if -1.45e-215 < z < 7.20000000000000014e-190

   1. Initial program 62.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.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. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 85.9

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

   5. Simplified 85.9%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 81.9

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

   8. Simplified 81.9%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

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

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

       6. mul-1-neg N/A

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

       7. lower-*.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 55.3

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

   11. Simplified 55.3%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+33}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-215}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{x \cdot z}{-y}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-76}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.36 \cdot 10^{+34}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.02 \cdot 10^{+65}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.36e+34)
   (+ x y)
   (if (<= z 2.02e+65) (- (- z) (/ (* x z) y)) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.36e+34) {
		tmp = x + y;
	} else if (z <= 2.02e+65) {
		tmp = -z - ((x * z) / y);
	} 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 <= (-1.36d+34)) then
        tmp = x + y
    else if (z <= 2.02d+65) then
        tmp = -z - ((x * z) / y)
    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 <= -1.36e+34) {
		tmp = x + y;
	} else if (z <= 2.02e+65) {
		tmp = -z - ((x * z) / y);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.36e+34:
		tmp = x + y
	elif z <= 2.02e+65:
		tmp = -z - ((x * z) / y)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.36e+34)
		tmp = Float64(x + y);
	elseif (z <= 2.02e+65)
		tmp = Float64(Float64(-z) - Float64(Float64(x * z) / y));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.36e+34)
		tmp = x + y;
	elseif (z <= 2.02e+65)
		tmp = -z - ((x * z) / y);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.36e+34], N[(x + y), $MachinePrecision], If[LessEqual[z, 2.02e+65], N[((-z) - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.36e34 or 2.0199999999999999e65 < 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. lower-+.f64 81.4

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

   5. Simplified 81.4%

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

   if -1.36e34 < z < 2.0199999999999999e65

   1. Initial program 79.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. lower--.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. lower-neg.f64 N/A

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

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

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

      19. lower-+.f64 75.6

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

   5. Simplified 75.6%

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

   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. lower-/.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. lower-*.f64 75.7

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

   8. Simplified 75.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.36 \cdot 10^{+34}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.02 \cdot 10^{+65}:\\ \;\;\;\;\left(-z\right) - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{+66}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.8e+90)
   (+ x y)
   (if (<= z 1.66e+66) (- (fma z (/ x y) z)) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.8e+90) {
		tmp = x + y;
	} else if (z <= 1.66e+66) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.8e+90)
		tmp = Float64(x + y);
	elseif (z <= 1.66e+66)
		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, -5.8e+90], N[(x + y), $MachinePrecision], If[LessEqual[z, 1.66e+66], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.8000000000000003e90 or 1.6600000000000001e66 < 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. lower-+.f64 85.2

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

   5. Simplified 85.2%

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

   if -5.8000000000000003e90 < z < 1.6600000000000001e66

   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. lower-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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+90}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{+66}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+33}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-76}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8e+33) (+ x y) (if (<= z 3.4e-76) (- z) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8e+33) {
		tmp = x + y;
	} else if (z <= 3.4e-76) {
		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 <= (-8d+33)) then
        tmp = x + y
    else if (z <= 3.4d-76) 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 <= -8e+33) {
		tmp = x + y;
	} else if (z <= 3.4e-76) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8e+33:
		tmp = x + y
	elif z <= 3.4e-76:
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8e+33)
		tmp = Float64(x + y);
	elseif (z <= 3.4e-76)
		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 <= -8e+33)
		tmp = x + y;
	elseif (z <= 3.4e-76)
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8e+33], N[(x + y), $MachinePrecision], If[LessEqual[z, 3.4e-76], (-z), N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.9999999999999996e33 or 3.3999999999999999e-76 < 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. lower-+.f64 72.8

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

   5. Simplified 72.8%

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

   if -7.9999999999999996e33 < z < 3.3999999999999999e-76

   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. lower-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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+33}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-76}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+103}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+41}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.4e+103) y (if (<= z 1.85e+41) (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.4e+103) {
		tmp = y;
	} else if (z <= 1.85e+41) {
		tmp = -z;
	} else {
		tmp = 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 <= (-3.4d+103)) then
        tmp = y
    else if (z <= 1.85d+41) then
        tmp = -z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.4e+103) {
		tmp = y;
	} else if (z <= 1.85e+41) {
		tmp = -z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.4e+103:
		tmp = y
	elif z <= 1.85e+41:
		tmp = -z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.4e+103)
		tmp = y;
	elseif (z <= 1.85e+41)
		tmp = Float64(-z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.4e+103)
		tmp = y;
	elseif (z <= 1.85e+41)
		tmp = -z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.4e+103], y, If[LessEqual[z, 1.85e+41], (-z), y]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3999999999999998e103 or 1.84999999999999991e41 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 40.7

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

   8. Simplified 40.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

         \[\leadsto y \cdot \color{blue}{1} \]

      4. *-rgt-identity 44.1

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

   10. Applied egg-rr 44.1%

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

   if -3.3999999999999998e103 < z < 1.84999999999999991e41

   1. Initial program 80.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. lower-neg.f64 45.2

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

   5. Simplified 45.2%

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

Alternative 8: 18.7% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 88.1%

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

3. Taylor expanded in x around 0

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

   1. *-inverses N/A

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

   2. div-sub N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower--.f64 51.7

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

5. Simplified 51.7%

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

6. Taylor expanded in y around 0

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

   1. lower-/.f64 21.8

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

8. Simplified 21.8%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. *-inverses N/A

      \[\leadsto y \cdot \color{blue}{1} \]

   4. *-rgt-identity 23.2

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

10. Applied egg-rr 23.2%

    \[\leadsto \color{blue}{y} \]
11. 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))))