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

Percentage Accurate: 87.9% → 99.7%

Time: 2.8s

Alternatives: 5

Speedup: N/A×

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: 87.9% 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: 99.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+74} \lor \neg \left(y \leq 10^{-48}\right):\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(-2, z, \left(-1 \cdot -2\right) \cdot y\right)} \cdot \left(-2 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{-1 \cdot \left(x + y\right)}{\frac{2 \cdot z - 2 \cdot y}{2 \cdot z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5e+74) (not (<= y 1e-48)))
   (* (/ (+ y x) (fma -2.0 z (* (* -1.0 -2.0) y))) (* -2.0 z))
   (* -1.0 (/ (* -1.0 (+ x y)) (/ (- (* 2.0 z) (* 2.0 y)) (* 2.0 z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e+74) || !(y <= 1e-48)) {
		tmp = ((y + x) / fma(-2.0, z, ((-1.0 * -2.0) * y))) * (-2.0 * z);
	} else {
		tmp = -1.0 * ((-1.0 * (x + y)) / (((2.0 * z) - (2.0 * y)) / (2.0 * z)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5e+74) || !(y <= 1e-48))
		tmp = Float64(Float64(Float64(y + x) / fma(-2.0, z, Float64(Float64(-1.0 * -2.0) * y))) * Float64(-2.0 * z));
	else
		tmp = Float64(-1.0 * Float64(Float64(-1.0 * Float64(x + y)) / Float64(Float64(Float64(2.0 * z) - Float64(2.0 * y)) / Float64(2.0 * z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5e+74], N[Not[LessEqual[y, 1e-48]], $MachinePrecision]], N[(N[(N[(y + x), $MachinePrecision] / N[(-2.0 * z + N[(N[(-1.0 * -2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * z), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(-1.0 * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 * z), $MachinePrecision] - N[(2.0 * y), $MachinePrecision]), $MachinePrecision] / N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999963e74 or 9.9999999999999997e-49 < y

   1. Initial program 72.9%

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

      1. lift--.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. mul-1-neg N/A

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

      6. frac-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 72.9

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

   4. Applied rewrites 72.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 99.9%

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

   if -4.99999999999999963e74 < y < 9.9999999999999997e-49

   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

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. mul-1-neg N/A

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

      6. frac-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \frac{x + y}{\color{blue}{\frac{2 \cdot \left(-1 \cdot z\right) - 2 \cdot \left(-y\right)}{2 \cdot \left(-1 \cdot z\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+74} \lor \neg \left(y \leq 10^{-48}\right):\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(-2, z, \left(-1 \cdot -2\right) \cdot y\right)} \cdot \left(-2 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{-1 \cdot \left(x + y\right)}{\frac{2 \cdot z - 2 \cdot y}{2 \cdot z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-19} \lor \neg \left(y \leq 10^{-48}\right):\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(-2, z, \left(-1 \cdot -2\right) \cdot y\right)} \cdot \left(-2 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1e-19) (not (<= y 1e-48)))
   (* (/ (+ y x) (fma -2.0 z (* (* -1.0 -2.0) y))) (* -2.0 z))
   (/ (+ x y) (- 1.0 (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e-19) || !(y <= 1e-48)) {
		tmp = ((y + x) / fma(-2.0, z, ((-1.0 * -2.0) * y))) * (-2.0 * z);
	} else {
		tmp = (x + y) / (1.0 - (y / z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1e-19) || !(y <= 1e-48))
		tmp = Float64(Float64(Float64(y + x) / fma(-2.0, z, Float64(Float64(-1.0 * -2.0) * y))) * Float64(-2.0 * z));
	else
		tmp = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1e-19], N[Not[LessEqual[y, 1e-48]], $MachinePrecision]], N[(N[(N[(y + x), $MachinePrecision] / N[(-2.0 * z + N[(N[(-1.0 * -2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * z), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999998e-20 or 9.9999999999999997e-49 < y

   1. Initial program 76.0%

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

      1. lift--.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. mul-1-neg N/A

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

      6. frac-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 76.0

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

   4. Applied rewrites 76.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 99.9%

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

   if -9.9999999999999998e-20 < y < 9.9999999999999997e-49

   1. Initial program 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-19} \lor \neg \left(y \leq 10^{-48}\right):\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(-2, z, \left(-1 \cdot -2\right) \cdot y\right)} \cdot \left(-2 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{y + x}{\mathsf{fma}\left(-2, z, \left(-1 \cdot -2\right) \cdot y\right)} \cdot \left(-2 \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (* (/ (+ y x) (fma -2.0 z (* (* -1.0 -2.0) y))) (* -2.0 z)))

C

double code(double x, double y, double z) {
	return ((y + x) / fma(-2.0, z, ((-1.0 * -2.0) * y))) * (-2.0 * z);
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y + x) / fma(-2.0, z, Float64(Float64(-1.0 * -2.0) * y))) * Float64(-2.0 * z))
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(y + x), $MachinePrecision] / N[(-2.0 * z + N[(N[(-1.0 * -2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.6%

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

   1. lift--.f64 N/A

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

   2. metadata-eval N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. mul-1-neg N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 85.6

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

4. Applied rewrites 85.6%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

6. Applied rewrites 91.1%

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

7. Final simplification 91.1%

   \[\leadsto \frac{y + x}{\mathsf{fma}\left(-2, z, \left(-1 \cdot -2\right) \cdot y\right)} \cdot \left(-2 \cdot z\right) \]
8. Add Preprocessing

Alternative 4: 46.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 64000000000:\\ \;\;\;\;\mathsf{fma}\left(-1 \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(x, \frac{z}{y}, \frac{z \cdot z}{y}\right), z \cdot x\right) - \left(z \cdot z\right) \cdot -1}{-1 \cdot y}, -1, -1 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 64000000000.0)
   (fma
    (*
     -1.0
     (/
      (- (fma z (fma x (/ z y) (/ (* z z) y)) (* z x)) (* (* z z) -1.0))
      (* -1.0 y)))
    -1.0
    (* -1.0 z))
   (* -1.0 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 64000000000.0) {
		tmp = fma((-1.0 * ((fma(z, fma(x, (z / y), ((z * z) / y)), (z * x)) - ((z * z) * -1.0)) / (-1.0 * y))), -1.0, (-1.0 * z));
	} else {
		tmp = -1.0 * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 64000000000.0)
		tmp = fma(Float64(-1.0 * Float64(Float64(fma(z, fma(x, Float64(z / y), Float64(Float64(z * z) / y)), Float64(z * x)) - Float64(Float64(z * z) * -1.0)) / Float64(-1.0 * y))), -1.0, Float64(-1.0 * z));
	else
		tmp = Float64(-1.0 * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 64000000000.0], N[(N[(-1.0 * N[(N[(N[(z * N[(x * N[(z / y), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] - N[(N[(z * z), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(-1.0 * z), $MachinePrecision]), $MachinePrecision], N[(-1.0 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.4e10

   1. Initial program 81.9%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 62.8%

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

   if 6.4e10 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 28.3

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

   5. Applied rewrites 28.3%

      \[\leadsto \color{blue}{-1 \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 64000000000:\\ \;\;\;\;\mathsf{fma}\left(-1 \cdot \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(x, \frac{z}{y}, \frac{z \cdot z}{y}\right), z \cdot x\right) - \left(z \cdot z\right) \cdot -1}{-1 \cdot y}, -1, -1 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ -1 \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* -1.0 z))

C

double code(double x, double y, double z) {
	return -1.0 * 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 = (-1.0d0) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(-1.0 * z), $MachinePrecision]

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 46.4

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

5. Applied rewrites 46.4%

   \[\leadsto \color{blue}{-1 \cdot z} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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))))