Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C

Percentage Accurate: 99.9% → 99.9%

Time: 4.4s

Alternatives: 13

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x + \sin y\right) + z \cdot \cos y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ x (sin y)) (* z (cos y))))

C

double code(double x, double y, double z) {
	return (x + sin(y)) + (z * cos(y));
}

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 + sin(y)) + (z * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x + Math.sin(y)) + (z * Math.cos(y));
}

Python

def code(x, y, z):
	return (x + math.sin(y)) + (z * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + sin(y)) + (z * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + \sin y\right) + z \cdot \cos y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ x (sin y)) (* z (cos y))))

C

double code(double x, double y, double z) {
	return (x + sin(y)) + (z * cos(y));
}

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 + sin(y)) + (z * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x + Math.sin(y)) + (z * Math.cos(y));
}

Python

def code(x, y, z):
	return (x + math.sin(y)) + (z * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + sin(y)) + (z * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos y, z, x\right) + \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (fma (cos y) z x) (sin y)))

C

double code(double x, double y, double z) {
	return fma(cos(y), z, x) + sin(y);
}

Julia

function code(x, y, z)
	return Float64(fma(cos(y), z, x) + sin(y))
end

Wolfram

code[x_, y_, z_] := N[(N[(N[Cos[y], $MachinePrecision] * z + x), $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x + \sin y\right) + z \cdot \cos y \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-+r+ N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lift-sin.f64 99.9

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

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x\right) + \sin y} \]
4. Add Preprocessing

Alternative 2: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos y, z, \sin y\right)\\ t_1 := z \cdot \cos y\\ t_2 := \left(x + \sin y\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -200000000:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left(z + y\right) + x\\ \mathbf{elif}\;t\_2 \leq 1000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (cos y) z (sin y)))
        (t_1 (* z (cos y)))
        (t_2 (+ (+ x (sin y)) t_1)))
   (if (<= t_2 -200000000.0)
     (fma (cos y) z x)
     (if (<= t_2 -1e-7)
       t_0
       (if (<= t_2 5e-9)
         (+ (+ z y) x)
         (if (<= t_2 1000000000.0) t_0 (+ t_1 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(cos(y), z, sin(y));
	double t_1 = z * cos(y);
	double t_2 = (x + sin(y)) + t_1;
	double tmp;
	if (t_2 <= -200000000.0) {
		tmp = fma(cos(y), z, x);
	} else if (t_2 <= -1e-7) {
		tmp = t_0;
	} else if (t_2 <= 5e-9) {
		tmp = (z + y) + x;
	} else if (t_2 <= 1000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(cos(y), z, sin(y))
	t_1 = Float64(z * cos(y))
	t_2 = Float64(Float64(x + sin(y)) + t_1)
	tmp = 0.0
	if (t_2 <= -200000000.0)
		tmp = fma(cos(y), z, x);
	elseif (t_2 <= -1e-7)
		tmp = t_0;
	elseif (t_2 <= 5e-9)
		tmp = Float64(Float64(z + y) + x);
	elseif (t_2 <= 1000000000.0)
		tmp = t_0;
	else
		tmp = Float64(t_1 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -200000000.0], N[(N[Cos[y], $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t$95$2, -1e-7], t$95$0, If[LessEqual[t$95$2, 5e-9], N[(N[(z + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 1000000000.0], t$95$0, N[(t$95$1 + x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -2e8

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + z \cdot \cos y \]
   3. Step-by-step derivation

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x} + z \cdot \cos y \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -2e8 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -9.9999999999999995e-8 or 5.0000000000000001e-9 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1e9

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\sin y + z \cdot \cos y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, \color{blue}{z}, \sin y\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos y, z, \sin y\right) \]

      5. lift-sin.f64 93.0

         \[\leadsto \mathsf{fma}\left(\cos y, z, \sin y\right) \]

   4. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, \sin y\right)} \]

   if -9.9999999999999995e-8 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 5.0000000000000001e-9

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto \left(z + y\right) + x \]

      4. lower-+.f64 100.0

         \[\leadsto \left(z + y\right) + x \]

   4. Applied rewrites 100.0%

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

   if 1e9 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + z \cdot \cos y \]
   3. Step-by-step derivation

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x} + z \cdot \cos y \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

Alternative 3: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y + x\\ t_1 := z \cdot \cos y\\ t_2 := \left(x + \sin y\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(\cos y, z, x\right)\\ \mathbf{elif}\;t\_2 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z\right) + x\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (sin y) x)) (t_1 (* z (cos y))) (t_2 (+ (+ x (sin y)) t_1)))
   (if (<= t_2 -1e+40)
     (fma (cos y) z x)
     (if (<= t_2 -0.02)
       t_0
       (if (<= t_2 2e-9)
         (+ (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y z) x)
         (if (<= t_2 1.0) t_0 (+ t_1 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) + x;
	double t_1 = z * cos(y);
	double t_2 = (x + sin(y)) + t_1;
	double tmp;
	if (t_2 <= -1e+40) {
		tmp = fma(cos(y), z, x);
	} else if (t_2 <= -0.02) {
		tmp = t_0;
	} else if (t_2 <= 2e-9) {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, z) + x;
	} else if (t_2 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) + x)
	t_1 = Float64(z * cos(y))
	t_2 = Float64(Float64(x + sin(y)) + t_1)
	tmp = 0.0
	if (t_2 <= -1e+40)
		tmp = fma(cos(y), z, x);
	elseif (t_2 <= -0.02)
		tmp = t_0;
	elseif (t_2 <= 2e-9)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, z) + x);
	elseif (t_2 <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(t_1 + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+40], N[(N[Cos[y], $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t$95$2, -0.02], t$95$0, If[LessEqual[t$95$2, 2e-9], N[(N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 1.0], t$95$0, N[(t$95$1 + x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -1.00000000000000003e40

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + z \cdot \cos y \]
   3. Step-by-step derivation

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{x} + z \cdot \cos y \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if -1.00000000000000003e40 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.0200000000000000004 or 2.00000000000000012e-9 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sin.f64 87.1

         \[\leadsto \sin y + x \]

   4. Applied rewrites 87.1%

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

   if -0.0200000000000000004 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 2.00000000000000012e-9

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \left(z + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right) + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(z + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right) + \color{blue}{x} \]

      3. +-commutative N/A

         \[\leadsto \left(y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) + z\right) + x \]

      4. *-commutative N/A

         \[\leadsto \left(\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) \cdot y + z\right) + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right), y, z\right) + x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) + 1, y, z\right) + x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) \cdot y + 1, y, z\right) + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y, y, 1\right), y, z\right) + x \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6} \cdot y + \frac{-1}{2} \cdot z, y, 1\right), y, z\right) + x \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, y, \frac{-1}{2} \cdot z\right), y, 1\right), y, z\right) + x \]

      11. lower-*.f64 99.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z\right) + x \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z\right) + x} \]

   if 1 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + z \cdot \cos y \]
   3. Step-by-step derivation

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{x} + z \cdot \cos y \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

Alternative 4: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y + x\\ t_1 := \left(x + \sin y\right) + z \cdot \cos y\\ t_2 := \mathsf{fma}\left(\cos y, z, x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+40}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z\right) + x\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (sin y) x))
        (t_1 (+ (+ x (sin y)) (* z (cos y))))
        (t_2 (fma (cos y) z x)))
   (if (<= t_1 -1e+40)
     t_2
     (if (<= t_1 -0.02)
       t_0
       (if (<= t_1 2e-9)
         (+ (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y z) x)
         (if (<= t_1 1.0) t_0 t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) + x;
	double t_1 = (x + sin(y)) + (z * cos(y));
	double t_2 = fma(cos(y), z, x);
	double tmp;
	if (t_1 <= -1e+40) {
		tmp = t_2;
	} else if (t_1 <= -0.02) {
		tmp = t_0;
	} else if (t_1 <= 2e-9) {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, z) + x;
	} else if (t_1 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) + x)
	t_1 = Float64(Float64(x + sin(y)) + Float64(z * cos(y)))
	t_2 = fma(cos(y), z, x)
	tmp = 0.0
	if (t_1 <= -1e+40)
		tmp = t_2;
	elseif (t_1 <= -0.02)
		tmp = t_0;
	elseif (t_1 <= 2e-9)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, z) + x);
	elseif (t_1 <= 1.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[y], $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+40], t$95$2, If[LessEqual[t$95$1, -0.02], t$95$0, If[LessEqual[t$95$1, 2e-9], N[(N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1.0], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -1.00000000000000003e40 or 1 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + z \cdot \cos y \]
   3. Step-by-step derivation

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{x} + z \cdot \cos y \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-cos.f64 99.6

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

   6. Applied rewrites 99.6%

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

   if -1.00000000000000003e40 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < -0.0200000000000000004 or 2.00000000000000012e-9 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 1

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sin.f64 87.1

         \[\leadsto \sin y + x \]

   4. Applied rewrites 87.1%

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

   if -0.0200000000000000004 < (+.f64 (+.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y))) < 2.00000000000000012e-9

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \left(z + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right) + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(z + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right) + \color{blue}{x} \]

      3. +-commutative N/A

         \[\leadsto \left(y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) + z\right) + x \]

      4. *-commutative N/A

         \[\leadsto \left(\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) \cdot y + z\right) + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right), y, z\right) + x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) + 1, y, z\right) + x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) \cdot y + 1, y, z\right) + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y, y, 1\right), y, z\right) + x \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6} \cdot y + \frac{-1}{2} \cdot z, y, 1\right), y, z\right) + x \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, y, \frac{-1}{2} \cdot z\right), y, 1\right), y, z\right) + x \]

      11. lower-*.f64 99.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z\right) + x \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z\right) + x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+182}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{+18}:\\ \;\;\;\;\sin y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cos y) z)))
   (if (<= z -4.2e+182)
     t_0
     (if (<= z -3.5e-9) (+ z x) (if (<= z 7.4e+18) (+ (sin y) x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) * z;
	double tmp;
	if (z <= -4.2e+182) {
		tmp = t_0;
	} else if (z <= -3.5e-9) {
		tmp = z + x;
	} else if (z <= 7.4e+18) {
		tmp = sin(y) + x;
	} 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 = cos(y) * z
    if (z <= (-4.2d+182)) then
        tmp = t_0
    else if (z <= (-3.5d-9)) then
        tmp = z + x
    else if (z <= 7.4d+18) then
        tmp = sin(y) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cos(y) * z;
	double tmp;
	if (z <= -4.2e+182) {
		tmp = t_0;
	} else if (z <= -3.5e-9) {
		tmp = z + x;
	} else if (z <= 7.4e+18) {
		tmp = Math.sin(y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cos(y) * z
	tmp = 0
	if z <= -4.2e+182:
		tmp = t_0
	elif z <= -3.5e-9:
		tmp = z + x
	elif z <= 7.4e+18:
		tmp = math.sin(y) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) * z)
	tmp = 0.0
	if (z <= -4.2e+182)
		tmp = t_0;
	elseif (z <= -3.5e-9)
		tmp = Float64(z + x);
	elseif (z <= 7.4e+18)
		tmp = Float64(sin(y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cos(y) * z;
	tmp = 0.0;
	if (z <= -4.2e+182)
		tmp = t_0;
	elseif (z <= -3.5e-9)
		tmp = z + x;
	elseif (z <= 7.4e+18)
		tmp = sin(y) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -4.2e+182], t$95$0, If[LessEqual[z, -3.5e-9], N[(z + x), $MachinePrecision], If[LessEqual[z, 7.4e+18], N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.1999999999999998e182 or 7.4e18 < z

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 81.3

         \[\leadsto \cos y \cdot z \]

   4. Applied rewrites 81.3%

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

   if -4.1999999999999998e182 < z < -3.4999999999999999e-9

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 71.1

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

   4. Applied rewrites 71.1%

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

   if -3.4999999999999999e-9 < z < 7.4e18

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sin.f64 90.6

         \[\leadsto \sin y + x \]

   4. Applied rewrites 90.6%

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

Alternative 6: 81.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y + x\\ \mathbf{if}\;y \leq -4000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.42:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (sin y) x)))
   (if (<= y -4000000000000.0)
     t_0
     (if (<= y 0.42)
       (+ (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y z) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = sin(y) + x;
	double tmp;
	if (y <= -4000000000000.0) {
		tmp = t_0;
	} else if (y <= 0.42) {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, z) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(sin(y) + x)
	tmp = 0.0
	if (y <= -4000000000000.0)
		tmp = t_0;
	elseif (y <= 0.42)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, z) + x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -4000000000000.0], t$95$0, If[LessEqual[y, 0.42], N[(N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + z), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4e12 or 0.419999999999999984 < y

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lift-sin.f64 62.9

         \[\leadsto \sin y + x \]

   4. Applied rewrites 62.9%

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

   if -4e12 < y < 0.419999999999999984

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \left(z + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right) + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(z + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right) + \color{blue}{x} \]

      3. +-commutative N/A

         \[\leadsto \left(y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) + z\right) + x \]

      4. *-commutative N/A

         \[\leadsto \left(\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) \cdot y + z\right) + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right), y, z\right) + x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) + 1, y, z\right) + x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) \cdot y + 1, y, z\right) + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y, y, 1\right), y, z\right) + x \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6} \cdot y + \frac{-1}{2} \cdot z, y, 1\right), y, z\right) + x \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, y, \frac{-1}{2} \cdot z\right), y, 1\right), y, z\right) + x \]

      11. lower-*.f64 98.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z\right) + x \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z\right) + x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+20}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 4.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z\right) + x\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.8e+20)
   (+ z x)
   (if (<= y 4.8)
     (+ (fma (fma (fma -0.16666666666666666 y (* -0.5 z)) y 1.0) y z) x)
     (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.8e+20) {
		tmp = z + x;
	} else if (y <= 4.8) {
		tmp = fma(fma(fma(-0.16666666666666666, y, (-0.5 * z)), y, 1.0), y, z) + x;
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.8e+20)
		tmp = Float64(z + x);
	elseif (y <= 4.8)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, y, Float64(-0.5 * z)), y, 1.0), y, z) + x);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.8e+20], N[(z + x), $MachinePrecision], If[LessEqual[y, 4.8], N[(N[(N[(N[(-0.16666666666666666 * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + z), $MachinePrecision] + x), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8e20 or 4.79999999999999982 < y

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 41.3

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

   4. Applied rewrites 41.3%

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

   if -7.8e20 < y < 4.79999999999999982

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

         \[\leadsto \left(z + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right) + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(z + y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right)\right) + \color{blue}{x} \]

      3. +-commutative N/A

         \[\leadsto \left(y \cdot \left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) + z\right) + x \]

      4. *-commutative N/A

         \[\leadsto \left(\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right)\right) \cdot y + z\right) + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(1 + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right), y, z\right) + x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) + 1, y, z\right) + x \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y\right) \cdot y + 1, y, z\right) + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot z + \frac{-1}{6} \cdot y, y, 1\right), y, z\right) + x \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6} \cdot y + \frac{-1}{2} \cdot z, y, 1\right), y, z\right) + x \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, y, \frac{-1}{2} \cdot z\right), y, 1\right), y, z\right) + x \]

      11. lower-*.f64 97.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z\right) + x \]

   4. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y, -0.5 \cdot z\right), y, 1\right), y, z\right) + x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5600000000000:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 4.8:\\ \;\;\;\;\mathsf{fma}\left(-\left(0.5 \cdot z - \frac{1}{y}\right) \cdot y, y, z + x\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5600000000000.0)
   (+ z x)
   (if (<= y 4.8) (fma (- (* (- (* 0.5 z) (/ 1.0 y)) y)) y (+ z x)) (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5600000000000.0) {
		tmp = z + x;
	} else if (y <= 4.8) {
		tmp = fma(-(((0.5 * z) - (1.0 / y)) * y), y, (z + x));
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5600000000000.0)
		tmp = Float64(z + x);
	elseif (y <= 4.8)
		tmp = fma(Float64(-Float64(Float64(Float64(0.5 * z) - Float64(1.0 / y)) * y)), y, Float64(z + x));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5600000000000.0], N[(z + x), $MachinePrecision], If[LessEqual[y, 4.8], N[((-N[(N[(N[(0.5 * z), $MachinePrecision] - N[(1.0 / y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]) * y + N[(z + x), $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.6e12 or 4.79999999999999982 < y

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 41.2

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

   4. Applied rewrites 41.2%

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

   if -5.6e12 < y < 4.79999999999999982

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, 1\right), y, z + x\right) \]

   4. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, 1\right), y, z + x\right)} \]

   5. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 98.1

         \[\leadsto \mathsf{fma}\left(-\left(0.5 \cdot z - \frac{1}{y}\right) \cdot y, y, z + x\right) \]

   7. Applied rewrites 98.1%

      \[\leadsto \mathsf{fma}\left(-\left(0.5 \cdot z - \frac{1}{y}\right) \cdot y, y, z + x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 70.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5600000000000:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 4.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, 1\right), y, z + x\right)\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5600000000000.0)
   (+ z x)
   (if (<= y 4.8) (fma (fma (* z y) -0.5 1.0) y (+ z x)) (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5600000000000.0) {
		tmp = z + x;
	} else if (y <= 4.8) {
		tmp = fma(fma((z * y), -0.5, 1.0), y, (z + x));
	} else {
		tmp = z + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5600000000000.0)
		tmp = Float64(z + x);
	elseif (y <= 4.8)
		tmp = fma(fma(Float64(z * y), -0.5, 1.0), y, Float64(z + x));
	else
		tmp = Float64(z + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5600000000000.0], N[(z + x), $MachinePrecision], If[LessEqual[y, 4.8], N[(N[(N[(z * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * y + N[(z + x), $MachinePrecision]), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.6e12 or 4.79999999999999982 < y

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 41.2

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

   4. Applied rewrites 41.2%

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

   if -5.6e12 < y < 4.79999999999999982

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, 1\right), y, z + x\right) \]

   4. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z \cdot y, -0.5, 1\right), y, z + x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5600000000000:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+86}:\\ \;\;\;\;\left(z + y\right) + x\\ \mathbf{else}:\\ \;\;\;\;z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5600000000000.0)
   (+ z x)
   (if (<= y 1.65e+86) (+ (+ z y) x) (+ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5600000000000.0) {
		tmp = z + x;
	} else if (y <= 1.65e+86) {
		tmp = (z + y) + x;
	} else {
		tmp = z + x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5600000000000.0d0)) then
        tmp = z + x
    else if (y <= 1.65d+86) then
        tmp = (z + y) + x
    else
        tmp = z + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5600000000000.0) {
		tmp = z + x;
	} else if (y <= 1.65e+86) {
		tmp = (z + y) + x;
	} else {
		tmp = z + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5600000000000.0:
		tmp = z + x
	elif y <= 1.65e+86:
		tmp = (z + y) + x
	else:
		tmp = z + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5600000000000.0)
		tmp = Float64(z + x);
	elseif (y <= 1.65e+86)
		tmp = Float64(Float64(z + y) + x);
	else
		tmp = Float64(z + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5600000000000.0)
		tmp = z + x;
	elseif (y <= 1.65e+86)
		tmp = (z + y) + x;
	else
		tmp = z + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5600000000000.0], N[(z + x), $MachinePrecision], If[LessEqual[y, 1.65e+86], N[(N[(z + y), $MachinePrecision] + x), $MachinePrecision], N[(z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.6e12 or 1.65e86 < y

   1. Initial program 99.8%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 41.6

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

   4. Applied rewrites 41.6%

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

   if -5.6e12 < y < 1.65e86

   1. Initial program 100.0%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto \left(z + y\right) + x \]

      4. lower-+.f64 91.0

         \[\leadsto \left(z + y\right) + x \]

   4. Applied rewrites 91.0%

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

Alternative 11: 66.4% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ z + x \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(z + x), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x + \sin y\right) + z \cdot \cos y \]

2. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 66.4

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

4. Applied rewrites 66.4%

   \[\leadsto \color{blue}{z + x} \]
5. Add Preprocessing

Alternative 12: 55.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-24}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.15e-51) x (if (<= x 3.2e-24) z x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.15e-51) {
		tmp = x;
	} else if (x <= 3.2e-24) {
		tmp = z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.15d-51)) then
        tmp = x
    else if (x <= 3.2d-24) then
        tmp = z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.15e-51) {
		tmp = x;
	} else if (x <= 3.2e-24) {
		tmp = z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.15e-51:
		tmp = x
	elif x <= 3.2e-24:
		tmp = z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.15e-51)
		tmp = x;
	elseif (x <= 3.2e-24)
		tmp = z;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.15e-51)
		tmp = x;
	elseif (x <= 3.2e-24)
		tmp = z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.15e-51], x, If[LessEqual[x, 3.2e-24], z, x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15000000000000001e-51 or 3.20000000000000012e-24 < x

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 69.1%

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

   if -1.15000000000000001e-51 < x < 3.20000000000000012e-24

   1. Initial program 99.9%

      \[\left(x + \sin y\right) + z \cdot \cos y \]

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 58.6

         \[\leadsto \cos y \cdot z \]

   4. Applied rewrites 58.6%

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

   5. Taylor expanded in y around 0

      \[\leadsto z \]
   6. Step-by-step derivation

   7. Applied rewrites 37.3%

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

Alternative 13: 42.5% accurate, 73.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.9%

   \[\left(x + \sin y\right) + z \cdot \cos y \]

2. Taylor expanded in x around inf

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

4. Applied rewrites 42.5%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, C"
  :precision binary64
  (+ (+ x (sin y)) (* z (cos y))))