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

Percentage Accurate: 88.2% → 99.1%

Time: 3.2s

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 88.2% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-237}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -5e-237) 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 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -5e-237) {
		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(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -5e-237)
		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[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-237], 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))) < -5.0000000000000002e-237 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 -5.0000000000000002e-237 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 18.5%

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

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 80.9

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

   8. Applied rewrites 80.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 93.8

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

   10. Applied rewrites 93.8%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. associate-/l* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-/.f64 94.7

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

   12. Applied rewrites 94.7%

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

Alternative 2: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{y}{z}, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (fma z (/ x y) z))))
   (if (<= y -6.2e+29) t_0 (if (<= y 1.9e-54) (+ (fma y (/ y z) 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 <= -6.2e+29) {
		tmp = t_0;
	} else if (y <= 1.9e-54) {
		tmp = fma(y, (y / z), 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 <= -6.2e+29)
		tmp = t_0;
	elseif (y <= 1.9e-54)
		tmp = Float64(fma(y, Float64(y / z), 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, -6.2e+29], t$95$0, If[LessEqual[y, 1.9e-54], N[(N[(y * N[(y / z), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.1999999999999998e29 or 1.9000000000000001e-54 < y

   1. Initial program 77.4%

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

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 60.8

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 69.0

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

   8. Applied rewrites 69.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 69.5

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

   10. Applied rewrites 69.5%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. associate-/l* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-/.f64 72.5

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

   12. Applied rewrites 72.5%

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

   if -6.1999999999999998e29 < y < 1.9000000000000001e-54

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.2%

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

Alternative 3: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-54}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (fma z (/ x y) z))))
   (if (<= y -6.2e+29) t_0 (if (<= y 1.9e-54) (+ 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 <= -6.2e+29) {
		tmp = t_0;
	} else if (y <= 1.9e-54) {
		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 <= -6.2e+29)
		tmp = t_0;
	elseif (y <= 1.9e-54)
		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, -6.2e+29], t$95$0, If[LessEqual[y, 1.9e-54], N[(y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.1999999999999998e29 or 1.9000000000000001e-54 < y

   1. Initial program 77.4%

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

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 60.8

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 69.0

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

   8. Applied rewrites 69.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 69.5

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

   10. Applied rewrites 69.5%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. associate-/l* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-/.f64 72.5

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

   12. Applied rewrites 72.5%

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

   if -6.1999999999999998e29 < y < 1.9000000000000001e-54

   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

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

      2. lower-+.f64 75.3

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

   5. Applied rewrites 75.3%

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

Alternative 4: 72.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-54}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (fma x (/ z y) z))))
   (if (<= y -6.6e+29) t_0 (if (<= y 1.9e-54) (+ y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -fma(x, (z / y), z);
	double tmp;
	if (y <= -6.6e+29) {
		tmp = t_0;
	} else if (y <= 1.9e-54) {
		tmp = y + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-fma(x, Float64(z / y), z))
	tmp = 0.0
	if (y <= -6.6e+29)
		tmp = t_0;
	elseif (y <= 1.9e-54)
		tmp = Float64(y + x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision])}, If[LessEqual[y, -6.6e+29], t$95$0, If[LessEqual[y, 1.9e-54], N[(y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.59999999999999968e29 or 1.9000000000000001e-54 < y

   1. Initial program 77.4%

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

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 60.8

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 69.0

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

   8. Applied rewrites 69.0%

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

   if -6.59999999999999968e29 < y < 1.9000000000000001e-54

   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

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

      2. lower-+.f64 75.3

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

   5. Applied rewrites 75.3%

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

Alternative 5: 66.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+36}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-21}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.22e+36) (- z) (if (<= y 2.8e-21) (+ y x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.22e+36) {
		tmp = -z;
	} else if (y <= 2.8e-21) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.22d+36)) then
        tmp = -z
    else if (y <= 2.8d-21) 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 <= -1.22e+36) {
		tmp = -z;
	} else if (y <= 2.8e-21) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.22e+36:
		tmp = -z
	elif y <= 2.8e-21:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.22e+36)
		tmp = Float64(-z);
	elseif (y <= 2.8e-21)
		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 <= -1.22e+36)
		tmp = -z;
	elseif (y <= 2.8e-21)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.22e+36], (-z), If[LessEqual[y, 2.8e-21], N[(y + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -1.21999999999999995e36 or 2.80000000000000004e-21 < y

   1. Initial program 75.9%

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

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

      2. lower-neg.f64 59.1

         \[\leadsto -z \]

   5. Applied rewrites 59.1%

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

   if -1.21999999999999995e36 < y < 2.80000000000000004e-21

   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

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

      2. lower-+.f64 73.7

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

   5. Applied rewrites 73.7%

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

Alternative 6: 58.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+16}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3e+16) (- z) (if (<= y 1.9e-54) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3e+16) {
		tmp = -z;
	} else if (y <= 1.9e-54) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3d+16)) then
        tmp = -z
    else if (y <= 1.9d-54) then
        tmp = 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 <= -3e+16) {
		tmp = -z;
	} else if (y <= 1.9e-54) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3e+16:
		tmp = -z
	elif y <= 1.9e-54:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3e+16)
		tmp = Float64(-z);
	elseif (y <= 1.9e-54)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3e+16)
		tmp = -z;
	elseif (y <= 1.9e-54)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3e+16], (-z), If[LessEqual[y, 1.9e-54], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -3e16 or 1.9000000000000001e-54 < y

   1. Initial program 77.9%

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

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

      2. lower-neg.f64 56.1

         \[\leadsto -z \]

   5. Applied rewrites 56.1%

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

   if -3e16 < y < 1.9000000000000001e-54

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 60.9%

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

Alternative 7: 40.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-179}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-131}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.6e-179) x (if (<= x 2.5e-131) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.6e-179) {
		tmp = x;
	} else if (x <= 2.5e-131) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.6d-179)) then
        tmp = x
    else if (x <= 2.5d-131) 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 <= -2.6e-179) {
		tmp = x;
	} else if (x <= 2.5e-131) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.6e-179:
		tmp = x
	elif x <= 2.5e-131:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.6e-179)
		tmp = x;
	elseif (x <= 2.5e-131)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.6e-179)
		tmp = x;
	elseif (x <= 2.5e-131)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.6e-179], x, If[LessEqual[x, 2.5e-131], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000005e-179 or 2.5000000000000002e-131 < x

   1. Initial program 87.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 41.4%

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

   if -2.60000000000000005e-179 < x < 2.5000000000000002e-131

   1. Initial program 89.3%

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

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 51.0

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto y + \color{blue}{\frac{{y}^{2}}{z}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{{y}^{2}}{z} + y \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{{y}^{2}}{z} + y \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{y}^{2}}{z} + y \]

      4. unpow2 N/A

         \[\leadsto \frac{y \cdot y}{z} + y \]

      5. lower-*.f64 35.3

         \[\leadsto \frac{y \cdot y}{z} + y \]

   8. Applied rewrites 35.3%

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

   9. Taylor expanded in y around 0

      \[\leadsto y \]
   10. Step-by-step derivation

   11. Applied rewrites 38.7%

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

Alternative 8: 35.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 88.2%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 35.0%

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 2025091 
(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))))