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

Percentage Accurate: 88.1% → 99.8%

Time: 5.8s

Alternatives: 5

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: 88.1% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-275}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))))
   (if (<= t_0 -1e-275) t_0 (if (<= t_0 0.0) (- (fma z (/ x y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -1e-275) {
		tmp = t_0;
	} 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(y + x) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -1e-275)
		tmp = t_0;
	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[(y + x), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-275], t$95$0, If[LessEqual[t$95$0, 0.0], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -9.99999999999999934e-276 or 0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

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

   if -9.99999999999999934e-276 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 6.3%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 82.0

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. unsub-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. associate-*r/ N/A

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

      10. div-sub N/A

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

      11. unsub-neg N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-out N/A

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

      14. lower-neg.f64 N/A

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

   8. Applied rewrites 82.5%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 82.5%

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

   13. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-275}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+105}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (fma z (/ x y) z))))
   (if (<= y -5.8e+69) t_0 (if (<= y 1.5e+105) (+ y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -fma(z, (x / y), z);
	double tmp;
	if (y <= -5.8e+69) {
		tmp = t_0;
	} else if (y <= 1.5e+105) {
		tmp = y + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-fma(z, Float64(x / y), z))
	tmp = 0.0
	if (y <= -5.8e+69)
		tmp = t_0;
	elseif (y <= 1.5e+105)
		tmp = Float64(y + x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision])}, If[LessEqual[y, -5.8e+69], t$95$0, If[LessEqual[y, 1.5e+105], N[(y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.7999999999999997e69 or 1.5e105 < y

   1. Initial program 75.2%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. unsub-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. associate-*r/ N/A

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

      10. div-sub N/A

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

      11. unsub-neg N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-out N/A

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

      14. lower-neg.f64 N/A

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

   8. Applied rewrites 75.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 76.1%

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

   13. Applied rewrites 80.8%

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

   if -5.7999999999999997e69 < y < 1.5e105

   1. Initial program 99.4%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 74.9

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

   8. Applied rewrites 74.9%

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

Alternative 3: 67.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+41}:\\ \;\;\;\;\frac{z}{z - y} \cdot y\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+110}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.4e+41) (* (/ z (- z y)) y) (if (<= y 2.6e+110) (+ y x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.4e+41) {
		tmp = (z / (z - y)) * y;
	} else if (y <= 2.6e+110) {
		tmp = y + x;
	} 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 <= (-3.4d+41)) then
        tmp = (z / (z - y)) * y
    else if (y <= 2.6d+110) then
        tmp = y + x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.4e+41) {
		tmp = (z / (z - y)) * y;
	} else if (y <= 2.6e+110) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.4e+41:
		tmp = (z / (z - y)) * y
	elif y <= 2.6e+110:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.4e+41)
		tmp = Float64(Float64(z / Float64(z - y)) * y);
	elseif (y <= 2.6e+110)
		tmp = Float64(y + x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.4e+41)
		tmp = (z / (z - y)) * y;
	elseif (y <= 2.6e+110)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.4e+41], N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2.6e+110], N[(y + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 3 regimes

2. if y < -3.39999999999999998e41

   1. Initial program 77.5%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.5%

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

   if -3.39999999999999998e41 < y < 2.6e110

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 74.7

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

   8. Applied rewrites 74.7%

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

   if 2.6e110 < y

   1. Initial program 68.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 77.1

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

   5. Applied rewrites 77.1%

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

Alternative 4: 68.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+71}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+110}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e+71) (- z) (if (<= y 2.6e+110) (+ y x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+71) {
		tmp = -z;
	} else if (y <= 2.6e+110) {
		tmp = y + x;
	} 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 <= (-1.4d+71)) then
        tmp = -z
    else if (y <= 2.6d+110) then
        tmp = y + x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+71) {
		tmp = -z;
	} else if (y <= 2.6e+110) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.4e+71:
		tmp = -z
	elif y <= 2.6e+110:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e+71)
		tmp = Float64(-z);
	elseif (y <= 2.6e+110)
		tmp = Float64(y + x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.4e+71)
		tmp = -z;
	elseif (y <= 2.6e+110)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.4e+71], (-z), If[LessEqual[y, 2.6e+110], N[(y + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -1.40000000000000001e71 or 2.6e110 < y

   1. Initial program 74.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 67.4

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

   5. Applied rewrites 67.4%

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

   if -1.40000000000000001e71 < y < 2.6e110

   1. Initial program 99.4%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 74.6

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

   8. Applied rewrites 74.6%

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

Alternative 5: 34.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return -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 = -z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 90.4%

   \[\frac{x + y}{1 - \frac{y}{z}} \]
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 32.1

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

5. Applied rewrites 32.1%

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

Developer Target 1: 94.0% 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 2024295 
(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))))