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

Percentage Accurate: 88.8% → 99.3%

Time: 10.5s

Alternatives: 7

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: 88.8% 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

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: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t\_0}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-237}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(\frac{x}{y}, z, z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{x - y} \cdot \frac{x - y}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ (+ x y) t_0)))
   (if (<= t_1 -2e-237)
     t_1
     (if (<= t_1 0.0)
       (- (fma (/ x y) z z))
       (* (/ (+ y x) (- x y)) (/ (- x y) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -2e-237) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = -fma((x / y), z, z);
	} else {
		tmp = ((y + x) / (x - y)) * ((x - y) / t_0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(Float64(x + y) / t_0)
	tmp = 0.0
	if (t_1 <= -2e-237)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(-fma(Float64(x / y), z, z));
	else
		tmp = Float64(Float64(Float64(y + x) / Float64(x - y)) * Float64(Float64(x - y) / t_0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-237], t$95$1, If[LessEqual[t$95$1, 0.0], (-N[(N[(x / y), $MachinePrecision] * z + z), $MachinePrecision]), N[(N[(N[(y + x), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision] * N[(N[(x - y), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -2e-237

   1. Initial program 99.8%

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

   if -2e-237 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 17.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. difference-of-squares N/A

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

      6. lift-+.f64 N/A

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

      7. times-frac N/A

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

      8. sub-div N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. sub-div N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 17.8

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

   4. Applied rewrites 17.8%

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

   5. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. div-add-rev N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg 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

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 100.0%

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

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. difference-of-squares N/A

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

      6. lift-+.f64 N/A

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

      7. times-frac N/A

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

      8. sub-div N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. sub-div N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{y + x}{x - y} \cdot \frac{x - y}{1 - \frac{y}{z}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-237} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{x}{y}, z, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-237) (not (<= t_0 0.0))) t_0 (- (fma (/ x y) z z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -2e-237) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -fma((x / y), z, z);
	}
	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 <= -2e-237) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(-fma(Float64(x / y), z, z));
	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[Or[LessEqual[t$95$0, -2e-237], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, (-N[(N[(x / y), $MachinePrecision] * z + z), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -2e-237 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

   if -2e-237 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 17.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. difference-of-squares N/A

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

      6. lift-+.f64 N/A

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

      7. times-frac N/A

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

      8. sub-div N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. sub-div N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 17.8

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

   4. Applied rewrites 17.8%

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

   5. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. div-add-rev N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg 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

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-237} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{x}{y}, z, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{fma}\left(\frac{x}{y}, z, z\right)\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-142}:\\ \;\;\;\;\left(y + x\right) \cdot \left(\frac{y}{z} + 1\right)\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (fma (/ x y) z z))))
   (if (<= y -4.2e-5)
     t_0
     (if (<= y -2.2e-142)
       (* (+ y x) (+ (/ y z) 1.0))
       (if (<= y 1.85e+93) (/ x (- 1.0 (/ y z))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = -fma((x / y), z, z);
	double tmp;
	if (y <= -4.2e-5) {
		tmp = t_0;
	} else if (y <= -2.2e-142) {
		tmp = (y + x) * ((y / z) + 1.0);
	} else if (y <= 1.85e+93) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-fma(Float64(x / y), z, z))
	tmp = 0.0
	if (y <= -4.2e-5)
		tmp = t_0;
	elseif (y <= -2.2e-142)
		tmp = Float64(Float64(y + x) * Float64(Float64(y / z) + 1.0));
	elseif (y <= 1.85e+93)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(N[(x / y), $MachinePrecision] * z + z), $MachinePrecision])}, If[LessEqual[y, -4.2e-5], t$95$0, If[LessEqual[y, -2.2e-142], N[(N[(y + x), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e+93], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.19999999999999977e-5 or 1.84999999999999994e93 < y

   1. Initial program 68.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. difference-of-squares N/A

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

      6. lift-+.f64 N/A

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

      7. times-frac N/A

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

      8. sub-div N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. sub-div N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 68.8

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

   4. Applied rewrites 68.8%

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

   5. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. div-add-rev N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg 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

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 81.2%

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

   if -4.19999999999999977e-5 < y < -2.20000000000000016e-142

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. difference-of-squares N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 73.4

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

   7. Applied rewrites 73.4%

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

   if -2.20000000000000016e-142 < y < 1.84999999999999994e93

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 79.0

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

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\frac{x}{1 - \frac{y}{z}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{x}{y}, z, z\right)\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-142}:\\ \;\;\;\;\left(y + x\right) \cdot \left(\frac{y}{z} + 1\right)\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{x}{y}, z, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-16} \lor \neg \left(z \leq 1.26 \cdot 10^{+54}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{z}{y}, x + z, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1e-16) (not (<= z 1.26e+54)))
   (+ x y)
   (- (fma (/ z y) (+ x z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-16) || !(z <= 1.26e+54)) {
		tmp = x + y;
	} else {
		tmp = -fma((z / y), (x + z), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1e-16) || !(z <= 1.26e+54))
		tmp = Float64(x + y);
	else
		tmp = Float64(-fma(Float64(z / y), Float64(x + z), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1e-16], N[Not[LessEqual[z, 1.26e+54]], $MachinePrecision]], N[(x + y), $MachinePrecision], (-N[(N[(z / y), $MachinePrecision] * N[(x + z), $MachinePrecision] + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z < -9.9999999999999998e-17 or 1.25999999999999995e54 < z

   1. Initial program 99.9%

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in z around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 79.3

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

   7. Applied rewrites 79.3%

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

   if -9.9999999999999998e-17 < z < 1.25999999999999995e54

   1. Initial program 76.1%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate--l- N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

      13. fp-cancel-sign-sub-inv N/A

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

      14. distribute-lft-out N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

   5. Applied rewrites 72.5%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-16} \lor \neg \left(z \leq 1.26 \cdot 10^{+54}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{z}{y}, x + z, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-16} \lor \neg \left(z \leq 1.26 \cdot 10^{+54}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{x}{y}, z, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1e-16) (not (<= z 1.26e+54))) (+ x y) (- (fma (/ x y) z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e-16) || !(z <= 1.26e+54)) {
		tmp = x + y;
	} else {
		tmp = -fma((x / y), z, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1e-16) || !(z <= 1.26e+54))
		tmp = Float64(x + y);
	else
		tmp = Float64(-fma(Float64(x / y), z, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1e-16], N[Not[LessEqual[z, 1.26e+54]], $MachinePrecision]], N[(x + y), $MachinePrecision], (-N[(N[(x / y), $MachinePrecision] * z + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z < -9.9999999999999998e-17 or 1.25999999999999995e54 < z

   1. Initial program 99.9%

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in z around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 79.3

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

   7. Applied rewrites 79.3%

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

   if -9.9999999999999998e-17 < z < 1.25999999999999995e54

   1. Initial program 76.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. difference-of-squares N/A

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

      6. lift-+.f64 N/A

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

      7. times-frac N/A

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

      8. sub-div N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 N/A

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

      15. sub-div N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. div-add-rev N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg 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

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   7. Applied rewrites 72.9%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 72.1%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-16} \lor \neg \left(z \leq 1.26 \cdot 10^{+54}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{x}{y}, z, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+65} \lor \neg \left(y \leq 2.8 \cdot 10^{+55}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.25e+65) (not (<= y 2.8e+55))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.25e+65) || !(y <= 2.8e+55)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	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 <= (-1.25d+65)) .or. (.not. (y <= 2.8d+55))) 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 ((y <= -1.25e+65) || !(y <= 2.8e+55)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.25e+65) or not (y <= 2.8e+55):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.25e+65) || !(y <= 2.8e+55))
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.25e+65) || ~((y <= 2.8e+55)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.25e+65], N[Not[LessEqual[y, 2.8e+55]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.24999999999999993e65 or 2.8000000000000001e55 < y

   1. Initial program 65.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 68.8

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

   5. Applied rewrites 68.8%

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

   if -1.24999999999999993e65 < y < 2.8000000000000001e55

   1. Initial program 99.3%

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in z around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 67.4

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

   7. Applied rewrites 67.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+65} \lor \neg \left(y \leq 2.8 \cdot 10^{+55}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.2% 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

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 = -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 86.1%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 35.9

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

5. Applied rewrites 35.9%

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

6. Final simplification 35.9%

   \[\leadsto -z \]
7. Add Preprocessing

Developer Target 1: 93.9% 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 2024359 
(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))))