Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B

Percentage Accurate: 99.7% → 99.7%

Time: 6.3s

Alternatives: 10

Speedup: 1.0×

• 21.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (- (- x y) (* 0.5 z)) 4.0) z))

C

double code(double x, double y, double z) {
	return (((x - y) - (0.5 * z)) * 4.0) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
end function

Java

public static double code(double x, double y, double z) {
	return (((x - y) - (0.5 * z)) * 4.0) / z;
}

Python

def code(x, y, z):
	return (((x - y) - (0.5 * z)) * 4.0) / z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x - y) - (0.5 * z)) * 4.0) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \]
4. Add Preprocessing

Alternative 2: 67.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 4}{z}\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ t_2 := \frac{-4 \cdot y}{z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+244}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x 4.0) z))
        (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z))
        (t_2 (/ (* -4.0 y) z)))
   (if (<= t_1 -1e+41)
     t_0
     (if (<= t_1 -1000000.0)
       t_2
       (if (<= t_1 -1.0)
         -2.0
         (if (<= t_1 2e+59) t_0 (if (<= t_1 1e+244) t_2 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double t_2 = (-4.0 * y) / z;
	double tmp;
	if (t_1 <= -1e+41) {
		tmp = t_0;
	} else if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+59) {
		tmp = t_0;
	} else if (t_1 <= 1e+244) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * 4.0d0) / z
    t_1 = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
    t_2 = ((-4.0d0) * y) / z
    if (t_1 <= (-1d+41)) then
        tmp = t_0
    else if (t_1 <= (-1000000.0d0)) then
        tmp = t_2
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t_1 <= 2d+59) then
        tmp = t_0
    else if (t_1 <= 1d+244) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double t_2 = (-4.0 * y) / z;
	double tmp;
	if (t_1 <= -1e+41) {
		tmp = t_0;
	} else if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+59) {
		tmp = t_0;
	} else if (t_1 <= 1e+244) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * 4.0) / z
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z
	t_2 = (-4.0 * y) / z
	tmp = 0
	if t_1 <= -1e+41:
		tmp = t_0
	elif t_1 <= -1000000.0:
		tmp = t_2
	elif t_1 <= -1.0:
		tmp = -2.0
	elif t_1 <= 2e+59:
		tmp = t_0
	elif t_1 <= 1e+244:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 4.0) / z)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	t_2 = Float64(Float64(-4.0 * y) / z)
	tmp = 0.0
	if (t_1 <= -1e+41)
		tmp = t_0;
	elseif (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+59)
		tmp = t_0;
	elseif (t_1 <= 1e+244)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * 4.0) / z;
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	t_2 = (-4.0 * y) / z;
	tmp = 0.0;
	if (t_1 <= -1e+41)
		tmp = t_0;
	elseif (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+59)
		tmp = t_0;
	elseif (t_1 <= 1e+244)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 4.0), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+41], t$95$0, If[LessEqual[t$95$1, -1000000.0], t$95$2, If[LessEqual[t$95$1, -1.0], -2.0, If[LessEqual[t$95$1, 2e+59], t$95$0, If[LessEqual[t$95$1, 1e+244], t$95$2, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1.00000000000000001e41 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 1.99999999999999994e59 or 1.00000000000000007e244 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot 4}}{z} \]

      2. lower-*.f64 61.8

         \[\leadsto \frac{\color{blue}{x \cdot 4}}{z} \]

   5. Applied rewrites 61.8%

      \[\leadsto \frac{\color{blue}{x \cdot 4}}{z} \]

   if -1.00000000000000001e41 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1e6 or 1.99999999999999994e59 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 1.00000000000000007e244

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 76.9

         \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   5. Applied rewrites 76.9%

      \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   if -1e6 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1 \cdot 10^{+41}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1000000:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq 10^{+244}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4}{z} \cdot x\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ t_2 := \frac{-4 \cdot y}{z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+244}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ 4.0 z) x))
        (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z))
        (t_2 (/ (* -4.0 y) z)))
   (if (<= t_1 -2e+69)
     t_0
     (if (<= t_1 -1000000.0)
       t_2
       (if (<= t_1 -1.0)
         -2.0
         (if (<= t_1 2e+59) t_0 (if (<= t_1 1e+244) t_2 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 / z) * x;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double t_2 = (-4.0 * y) / z;
	double tmp;
	if (t_1 <= -2e+69) {
		tmp = t_0;
	} else if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+59) {
		tmp = t_0;
	} else if (t_1 <= 1e+244) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (4.0d0 / z) * x
    t_1 = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
    t_2 = ((-4.0d0) * y) / z
    if (t_1 <= (-2d+69)) then
        tmp = t_0
    else if (t_1 <= (-1000000.0d0)) then
        tmp = t_2
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t_1 <= 2d+59) then
        tmp = t_0
    else if (t_1 <= 1d+244) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 / z) * x;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double t_2 = (-4.0 * y) / z;
	double tmp;
	if (t_1 <= -2e+69) {
		tmp = t_0;
	} else if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+59) {
		tmp = t_0;
	} else if (t_1 <= 1e+244) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 / z) * x
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z
	t_2 = (-4.0 * y) / z
	tmp = 0
	if t_1 <= -2e+69:
		tmp = t_0
	elif t_1 <= -1000000.0:
		tmp = t_2
	elif t_1 <= -1.0:
		tmp = -2.0
	elif t_1 <= 2e+59:
		tmp = t_0
	elif t_1 <= 1e+244:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 / z) * x)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	t_2 = Float64(Float64(-4.0 * y) / z)
	tmp = 0.0
	if (t_1 <= -2e+69)
		tmp = t_0;
	elseif (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+59)
		tmp = t_0;
	elseif (t_1 <= 1e+244)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 / z) * x;
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	t_2 = (-4.0 * y) / z;
	tmp = 0.0;
	if (t_1 <= -2e+69)
		tmp = t_0;
	elseif (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+59)
		tmp = t_0;
	elseif (t_1 <= 1e+244)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 / z), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+69], t$95$0, If[LessEqual[t$95$1, -1000000.0], t$95$2, If[LessEqual[t$95$1, -1.0], -2.0, If[LessEqual[t$95$1, 2e+59], t$95$0, If[LessEqual[t$95$1, 1e+244], t$95$2, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -2.0000000000000001e69 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 1.99999999999999994e59 or 1.00000000000000007e244 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{\color{blue}{1 \cdot x}}{z} \]

      2. associate-*l/ N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{1}{z} \cdot x\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot x} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{4 \cdot 1}{z}} \cdot x \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{4}}{z} \cdot x \]

      7. lower-/.f64 62.3

         \[\leadsto \color{blue}{\frac{4}{z}} \cdot x \]

   5. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot x} \]

   if -2.0000000000000001e69 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1e6 or 1.99999999999999994e59 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 1.00000000000000007e244

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 73.0

         \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   5. Applied rewrites 73.0%

      \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   if -1e6 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.8%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -2 \cdot 10^{+69}:\\ \;\;\;\;\frac{4}{z} \cdot x\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1000000:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\frac{4}{z} \cdot x\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq 10^{+244}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot 4}{z}\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) 4.0) z)) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -4e+15) t_0 (if (<= t_1 -1.0) (fma (/ y z) -4.0 -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - y) * 4.0) / z;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = fma((y / z), -4.0, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - y) * 4.0) / z)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -4e+15)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = fma(Float64(y / z), -4.0, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+15], t$95$0, If[LessEqual[t$95$1, -1.0], N[(N[(y / z), $MachinePrecision] * -4.0 + -2.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -4e15 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower--.f64 99.9

         \[\leadsto \frac{4 \cdot \color{blue}{\left(x - y\right)}}{z} \]

   5. Applied rewrites 99.9%

      \[\leadsto \frac{4 \cdot \color{blue}{\left(x - y\right)}}{z} \]

   if -4e15 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} - \frac{\frac{1}{2} \cdot z}{z}\right)} \]

      2. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)} \]

      4. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{\color{blue}{1 \cdot x}}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{1}{z} \cdot x\right)} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot x} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{z}{z}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{1}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + \color{blue}{-2} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \frac{1}{z}, x, -2\right)} \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4 \cdot 1}{z}}, x, -2\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{4}}{z}, x, -2\right) \]

      15. lower-/.f64 95.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4}{z}}, x, -2\right) \]

   5. Applied rewrites 95.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{z}, x, -2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.5%

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, \color{blue}{4}, -2\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]
   9. Step-by-step derivation

      1. --rgt-identity N/A

         \[\leadsto -4 \cdot \frac{\color{blue}{\left(y - 0\right)} + \frac{1}{2} \cdot z}{z} \]

      2. associate--r- N/A

         \[\leadsto -4 \cdot \frac{\color{blue}{y - \left(0 - \frac{1}{2} \cdot z\right)}}{z} \]

      3. neg-sub0 N/A

         \[\leadsto -4 \cdot \frac{y - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot z\right)\right)}}{z} \]

      4. div-sub N/A

         \[\leadsto -4 \cdot \color{blue}{\left(\frac{y}{z} - \frac{\mathsf{neg}\left(\frac{1}{2} \cdot z\right)}{z}\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot z}}{z}\right) \]

      6. metadata-eval N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{\color{blue}{\frac{-1}{2}} \cdot z}{z}\right) \]

      7. associate-/l* N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \color{blue}{\frac{-1}{2} \cdot \frac{z}{z}}\right) \]

      8. *-inverses N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{-1}{2} \cdot \color{blue}{1}\right) \]

      9. metadata-eval N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \color{blue}{\frac{-1}{2}}\right) \]

      10. sub-neg N/A

          \[\leadsto -4 \cdot \color{blue}{\left(\frac{y}{z} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto -4 \cdot \left(\frac{y}{z} + \color{blue}{\frac{1}{2}}\right) \]

      12. +-commutative N/A

          \[\leadsto -4 \cdot \color{blue}{\left(\frac{1}{2} + \frac{y}{z}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(\frac{1}{2} + \frac{y}{z}\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(4 \cdot \left(\frac{1}{2} + \frac{y}{z}\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto \mathsf{neg}\left(4 \cdot \color{blue}{\left(\frac{y}{z} + \frac{1}{2}\right)}\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{y}{z} + 4 \cdot \frac{1}{2}\right)}\right) \]

      17. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\left(4 \cdot \frac{\color{blue}{1 \cdot y}}{z} + 4 \cdot \frac{1}{2}\right)\right) \]

      18. associate-*l/ N/A

          \[\leadsto \mathsf{neg}\left(\left(4 \cdot \color{blue}{\left(\frac{1}{z} \cdot y\right)} + 4 \cdot \frac{1}{2}\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot y} + 4 \cdot \frac{1}{2}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\color{blue}{y \cdot \left(4 \cdot \frac{1}{z}\right)} + 4 \cdot \frac{1}{2}\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(4 \cdot \frac{1}{z}\right) + \color{blue}{2}\right)\right) \]

   10. Applied rewrites 99.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.2%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot 4}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot 4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4}{z} \cdot x\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ 4.0 z) x)) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -4e+15) t_0 (if (<= t_1 -1.0) -2.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 / z) * x;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (4.0d0 / z) * x
    t_1 = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
    if (t_1 <= (-4d+15)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 / z) * x;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -4e+15) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 / z) * x
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z
	tmp = 0
	if t_1 <= -4e+15:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 / z) * x)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -4e+15)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 / z) * x;
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	tmp = 0.0;
	if (t_1 <= -4e+15)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 / z), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+15], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -4e15 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{\color{blue}{1 \cdot x}}{z} \]

      2. associate-*l/ N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{1}{z} \cdot x\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot x} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{4 \cdot 1}{z}} \cdot x \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{4}}{z} \cdot x \]

      7. lower-/.f64 53.3

         \[\leadsto \color{blue}{\frac{4}{z}} \cdot x \]

   5. Applied rewrites 53.3%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot x} \]

   if -4e15 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.7%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\frac{4}{z} \cdot x\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) 4.0 -2.0)))
   (if (<= x -6.8e-57) t_0 (if (<= x 2.8e+37) (fma (/ y z) -4.0 -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), 4.0, -2.0);
	double tmp;
	if (x <= -6.8e-57) {
		tmp = t_0;
	} else if (x <= 2.8e+37) {
		tmp = fma((y / z), -4.0, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / z), 4.0, -2.0)
	tmp = 0.0
	if (x <= -6.8e-57)
		tmp = t_0;
	elseif (x <= 2.8e+37)
		tmp = fma(Float64(y / z), -4.0, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]}, If[LessEqual[x, -6.8e-57], t$95$0, If[LessEqual[x, 2.8e+37], N[(N[(y / z), $MachinePrecision] * -4.0 + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.80000000000000032e-57 or 2.7999999999999998e37 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} - \frac{\frac{1}{2} \cdot z}{z}\right)} \]

      2. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)} \]

      4. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{\color{blue}{1 \cdot x}}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{1}{z} \cdot x\right)} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot x} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{z}{z}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{1}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + \color{blue}{-2} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \frac{1}{z}, x, -2\right)} \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4 \cdot 1}{z}}, x, -2\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{4}}{z}, x, -2\right) \]

      15. lower-/.f64 86.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4}{z}}, x, -2\right) \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{z}, x, -2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.3%

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, \color{blue}{4}, -2\right) \]

   if -6.80000000000000032e-57 < x < 2.7999999999999998e37

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} - \frac{\frac{1}{2} \cdot z}{z}\right)} \]

      2. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)} \]

      4. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{\color{blue}{1 \cdot x}}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{1}{z} \cdot x\right)} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot x} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{z}{z}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{1}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + \color{blue}{-2} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \frac{1}{z}, x, -2\right)} \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4 \cdot 1}{z}}, x, -2\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{4}}{z}, x, -2\right) \]

      15. lower-/.f64 51.9

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4}{z}}, x, -2\right) \]

   5. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{z}, x, -2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, \color{blue}{4}, -2\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]
   9. Step-by-step derivation

      1. --rgt-identity N/A

         \[\leadsto -4 \cdot \frac{\color{blue}{\left(y - 0\right)} + \frac{1}{2} \cdot z}{z} \]

      2. associate--r- N/A

         \[\leadsto -4 \cdot \frac{\color{blue}{y - \left(0 - \frac{1}{2} \cdot z\right)}}{z} \]

      3. neg-sub0 N/A

         \[\leadsto -4 \cdot \frac{y - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot z\right)\right)}}{z} \]

      4. div-sub N/A

         \[\leadsto -4 \cdot \color{blue}{\left(\frac{y}{z} - \frac{\mathsf{neg}\left(\frac{1}{2} \cdot z\right)}{z}\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot z}}{z}\right) \]

      6. metadata-eval N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{\color{blue}{\frac{-1}{2}} \cdot z}{z}\right) \]

      7. associate-/l* N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \color{blue}{\frac{-1}{2} \cdot \frac{z}{z}}\right) \]

      8. *-inverses N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{-1}{2} \cdot \color{blue}{1}\right) \]

      9. metadata-eval N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \color{blue}{\frac{-1}{2}}\right) \]

      10. sub-neg N/A

          \[\leadsto -4 \cdot \color{blue}{\left(\frac{y}{z} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto -4 \cdot \left(\frac{y}{z} + \color{blue}{\frac{1}{2}}\right) \]

      12. +-commutative N/A

          \[\leadsto -4 \cdot \color{blue}{\left(\frac{1}{2} + \frac{y}{z}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(\frac{1}{2} + \frac{y}{z}\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(4 \cdot \left(\frac{1}{2} + \frac{y}{z}\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto \mathsf{neg}\left(4 \cdot \color{blue}{\left(\frac{y}{z} + \frac{1}{2}\right)}\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{y}{z} + 4 \cdot \frac{1}{2}\right)}\right) \]

      17. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\left(4 \cdot \frac{\color{blue}{1 \cdot y}}{z} + 4 \cdot \frac{1}{2}\right)\right) \]

      18. associate-*l/ N/A

          \[\leadsto \mathsf{neg}\left(\left(4 \cdot \color{blue}{\left(\frac{1}{z} \cdot y\right)} + 4 \cdot \frac{1}{2}\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot y} + 4 \cdot \frac{1}{2}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\color{blue}{y \cdot \left(4 \cdot \frac{1}{z}\right)} + 4 \cdot \frac{1}{2}\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(4 \cdot \frac{1}{z}\right) + \color{blue}{2}\right)\right) \]

   10. Applied rewrites 91.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 91.7%

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

Alternative 7: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) 4.0 -2.0)))
   (if (<= x -6.8e-57) t_0 (if (<= x 2.8e+37) (fma (/ -4.0 z) y -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), 4.0, -2.0);
	double tmp;
	if (x <= -6.8e-57) {
		tmp = t_0;
	} else if (x <= 2.8e+37) {
		tmp = fma((-4.0 / z), y, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / z), 4.0, -2.0)
	tmp = 0.0
	if (x <= -6.8e-57)
		tmp = t_0;
	elseif (x <= 2.8e+37)
		tmp = fma(Float64(-4.0 / z), y, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]}, If[LessEqual[x, -6.8e-57], t$95$0, If[LessEqual[x, 2.8e+37], N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.80000000000000032e-57 or 2.7999999999999998e37 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} - \frac{\frac{1}{2} \cdot z}{z}\right)} \]

      2. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)} \]

      4. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{\color{blue}{1 \cdot x}}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{1}{z} \cdot x\right)} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot x} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{z}{z}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{1}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + \color{blue}{-2} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \frac{1}{z}, x, -2\right)} \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4 \cdot 1}{z}}, x, -2\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{4}}{z}, x, -2\right) \]

      15. lower-/.f64 86.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4}{z}}, x, -2\right) \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{z}, x, -2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.3%

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, \color{blue}{4}, -2\right) \]

   if -6.80000000000000032e-57 < x < 2.7999999999999998e37

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} - \frac{\frac{1}{2} \cdot z}{z}\right)} \]

      2. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)} \]

      4. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{\color{blue}{1 \cdot x}}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{1}{z} \cdot x\right)} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot x} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{z}{z}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{1}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + \color{blue}{-2} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \frac{1}{z}, x, -2\right)} \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4 \cdot 1}{z}}, x, -2\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{4}}{z}, x, -2\right) \]

      15. lower-/.f64 51.9

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4}{z}}, x, -2\right) \]

   5. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{z}, x, -2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, \color{blue}{4}, -2\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]
   9. Step-by-step derivation

      1. --rgt-identity N/A

         \[\leadsto -4 \cdot \frac{\color{blue}{\left(y - 0\right)} + \frac{1}{2} \cdot z}{z} \]

      2. associate--r- N/A

         \[\leadsto -4 \cdot \frac{\color{blue}{y - \left(0 - \frac{1}{2} \cdot z\right)}}{z} \]

      3. neg-sub0 N/A

         \[\leadsto -4 \cdot \frac{y - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot z\right)\right)}}{z} \]

      4. div-sub N/A

         \[\leadsto -4 \cdot \color{blue}{\left(\frac{y}{z} - \frac{\mathsf{neg}\left(\frac{1}{2} \cdot z\right)}{z}\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot z}}{z}\right) \]

      6. metadata-eval N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{\color{blue}{\frac{-1}{2}} \cdot z}{z}\right) \]

      7. associate-/l* N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \color{blue}{\frac{-1}{2} \cdot \frac{z}{z}}\right) \]

      8. *-inverses N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{-1}{2} \cdot \color{blue}{1}\right) \]

      9. metadata-eval N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \color{blue}{\frac{-1}{2}}\right) \]

      10. sub-neg N/A

          \[\leadsto -4 \cdot \color{blue}{\left(\frac{y}{z} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto -4 \cdot \left(\frac{y}{z} + \color{blue}{\frac{1}{2}}\right) \]

      12. +-commutative N/A

          \[\leadsto -4 \cdot \color{blue}{\left(\frac{1}{2} + \frac{y}{z}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(\frac{1}{2} + \frac{y}{z}\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(4 \cdot \left(\frac{1}{2} + \frac{y}{z}\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto \mathsf{neg}\left(4 \cdot \color{blue}{\left(\frac{y}{z} + \frac{1}{2}\right)}\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{y}{z} + 4 \cdot \frac{1}{2}\right)}\right) \]

      17. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\left(4 \cdot \frac{\color{blue}{1 \cdot y}}{z} + 4 \cdot \frac{1}{2}\right)\right) \]

      18. associate-*l/ N/A

          \[\leadsto \mathsf{neg}\left(\left(4 \cdot \color{blue}{\left(\frac{1}{z} \cdot y\right)} + 4 \cdot \frac{1}{2}\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot y} + 4 \cdot \frac{1}{2}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\color{blue}{y \cdot \left(4 \cdot \frac{1}{z}\right)} + 4 \cdot \frac{1}{2}\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(4 \cdot \frac{1}{z}\right) + \color{blue}{2}\right)\right) \]

   10. Applied rewrites 91.6%

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

Alternative 8: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ 4.0 z) x -2.0)))
   (if (<= x -6.8e-57) t_0 (if (<= x 2.8e+37) (fma (/ -4.0 z) y -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((4.0 / z), x, -2.0);
	double tmp;
	if (x <= -6.8e-57) {
		tmp = t_0;
	} else if (x <= 2.8e+37) {
		tmp = fma((-4.0 / z), y, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(4.0 / z), x, -2.0)
	tmp = 0.0
	if (x <= -6.8e-57)
		tmp = t_0;
	elseif (x <= 2.8e+37)
		tmp = fma(Float64(-4.0 / z), y, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 / z), $MachinePrecision] * x + -2.0), $MachinePrecision]}, If[LessEqual[x, -6.8e-57], t$95$0, If[LessEqual[x, 2.8e+37], N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.80000000000000032e-57 or 2.7999999999999998e37 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} - \frac{\frac{1}{2} \cdot z}{z}\right)} \]

      2. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)} \]

      4. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{\color{blue}{1 \cdot x}}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{1}{z} \cdot x\right)} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot x} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{z}{z}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{1}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + \color{blue}{-2} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \frac{1}{z}, x, -2\right)} \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4 \cdot 1}{z}}, x, -2\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{4}}{z}, x, -2\right) \]

      15. lower-/.f64 86.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4}{z}}, x, -2\right) \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{z}, x, -2\right)} \]

   if -6.80000000000000032e-57 < x < 2.7999999999999998e37

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} - \frac{\frac{1}{2} \cdot z}{z}\right)} \]

      2. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)} \]

      4. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{\color{blue}{1 \cdot x}}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{1}{z} \cdot x\right)} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot x} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{z}{z}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{1}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + \color{blue}{-2} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \frac{1}{z}, x, -2\right)} \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4 \cdot 1}{z}}, x, -2\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{4}}{z}, x, -2\right) \]

      15. lower-/.f64 51.9

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4}{z}}, x, -2\right) \]

   5. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{z}, x, -2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, \color{blue}{4}, -2\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]
   9. Step-by-step derivation

      1. --rgt-identity N/A

         \[\leadsto -4 \cdot \frac{\color{blue}{\left(y - 0\right)} + \frac{1}{2} \cdot z}{z} \]

      2. associate--r- N/A

         \[\leadsto -4 \cdot \frac{\color{blue}{y - \left(0 - \frac{1}{2} \cdot z\right)}}{z} \]

      3. neg-sub0 N/A

         \[\leadsto -4 \cdot \frac{y - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot z\right)\right)}}{z} \]

      4. div-sub N/A

         \[\leadsto -4 \cdot \color{blue}{\left(\frac{y}{z} - \frac{\mathsf{neg}\left(\frac{1}{2} \cdot z\right)}{z}\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot z}}{z}\right) \]

      6. metadata-eval N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{\color{blue}{\frac{-1}{2}} \cdot z}{z}\right) \]

      7. associate-/l* N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \color{blue}{\frac{-1}{2} \cdot \frac{z}{z}}\right) \]

      8. *-inverses N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{-1}{2} \cdot \color{blue}{1}\right) \]

      9. metadata-eval N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \color{blue}{\frac{-1}{2}}\right) \]

      10. sub-neg N/A

          \[\leadsto -4 \cdot \color{blue}{\left(\frac{y}{z} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto -4 \cdot \left(\frac{y}{z} + \color{blue}{\frac{1}{2}}\right) \]

      12. +-commutative N/A

          \[\leadsto -4 \cdot \color{blue}{\left(\frac{1}{2} + \frac{y}{z}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(\frac{1}{2} + \frac{y}{z}\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(4 \cdot \left(\frac{1}{2} + \frac{y}{z}\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto \mathsf{neg}\left(4 \cdot \color{blue}{\left(\frac{y}{z} + \frac{1}{2}\right)}\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{y}{z} + 4 \cdot \frac{1}{2}\right)}\right) \]

      17. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\left(4 \cdot \frac{\color{blue}{1 \cdot y}}{z} + 4 \cdot \frac{1}{2}\right)\right) \]

      18. associate-*l/ N/A

          \[\leadsto \mathsf{neg}\left(\left(4 \cdot \color{blue}{\left(\frac{1}{z} \cdot y\right)} + 4 \cdot \frac{1}{2}\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot y} + 4 \cdot \frac{1}{2}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\color{blue}{y \cdot \left(4 \cdot \frac{1}{z}\right)} + 4 \cdot \frac{1}{2}\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(4 \cdot \frac{1}{z}\right) + \color{blue}{2}\right)\right) \]

   10. Applied rewrites 91.6%

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

Alternative 9: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 4}{z}\\ \mathbf{if}\;x \leq -3 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x 4.0) z)))
   (if (<= x -3e+88) t_0 (if (<= x 1.2e+38) (fma (/ -4.0 z) y -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double tmp;
	if (x <= -3e+88) {
		tmp = t_0;
	} else if (x <= 1.2e+38) {
		tmp = fma((-4.0 / z), y, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 4.0) / z)
	tmp = 0.0
	if (x <= -3e+88)
		tmp = t_0;
	elseif (x <= 1.2e+38)
		tmp = fma(Float64(-4.0 / z), y, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[x, -3e+88], t$95$0, If[LessEqual[x, 1.2e+38], N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.00000000000000005e88 or 1.20000000000000009e38 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot 4}}{z} \]

      2. lower-*.f64 78.2

         \[\leadsto \frac{\color{blue}{x \cdot 4}}{z} \]

   5. Applied rewrites 78.2%

      \[\leadsto \frac{\color{blue}{x \cdot 4}}{z} \]

   if -3.00000000000000005e88 < x < 1.20000000000000009e38

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} - \frac{\frac{1}{2} \cdot z}{z}\right)} \]

      2. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)} \]

      4. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{\color{blue}{1 \cdot x}}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{1}{z} \cdot x\right)} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot x} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{z}{z}}\right)\right) \]

      8. *-inverses N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{1}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot x + \color{blue}{-2} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(4 \cdot \frac{1}{z}, x, -2\right)} \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4 \cdot 1}{z}}, x, -2\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{4}}{z}, x, -2\right) \]

      15. lower-/.f64 57.0

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{4}{z}}, x, -2\right) \]

   5. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{z}, x, -2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 57.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, \color{blue}{4}, -2\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]
   9. Step-by-step derivation

      1. --rgt-identity N/A

         \[\leadsto -4 \cdot \frac{\color{blue}{\left(y - 0\right)} + \frac{1}{2} \cdot z}{z} \]

      2. associate--r- N/A

         \[\leadsto -4 \cdot \frac{\color{blue}{y - \left(0 - \frac{1}{2} \cdot z\right)}}{z} \]

      3. neg-sub0 N/A

         \[\leadsto -4 \cdot \frac{y - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot z\right)\right)}}{z} \]

      4. div-sub N/A

         \[\leadsto -4 \cdot \color{blue}{\left(\frac{y}{z} - \frac{\mathsf{neg}\left(\frac{1}{2} \cdot z\right)}{z}\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot z}}{z}\right) \]

      6. metadata-eval N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{\color{blue}{\frac{-1}{2}} \cdot z}{z}\right) \]

      7. associate-/l* N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \color{blue}{\frac{-1}{2} \cdot \frac{z}{z}}\right) \]

      8. *-inverses N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \frac{-1}{2} \cdot \color{blue}{1}\right) \]

      9. metadata-eval N/A

         \[\leadsto -4 \cdot \left(\frac{y}{z} - \color{blue}{\frac{-1}{2}}\right) \]

      10. sub-neg N/A

          \[\leadsto -4 \cdot \color{blue}{\left(\frac{y}{z} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto -4 \cdot \left(\frac{y}{z} + \color{blue}{\frac{1}{2}}\right) \]

      12. +-commutative N/A

          \[\leadsto -4 \cdot \color{blue}{\left(\frac{1}{2} + \frac{y}{z}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(\frac{1}{2} + \frac{y}{z}\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(4 \cdot \left(\frac{1}{2} + \frac{y}{z}\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto \mathsf{neg}\left(4 \cdot \color{blue}{\left(\frac{y}{z} + \frac{1}{2}\right)}\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{y}{z} + 4 \cdot \frac{1}{2}\right)}\right) \]

      17. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\left(4 \cdot \frac{\color{blue}{1 \cdot y}}{z} + 4 \cdot \frac{1}{2}\right)\right) \]

      18. associate-*l/ N/A

          \[\leadsto \mathsf{neg}\left(\left(4 \cdot \color{blue}{\left(\frac{1}{z} \cdot y\right)} + 4 \cdot \frac{1}{2}\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(4 \cdot \frac{1}{z}\right) \cdot y} + 4 \cdot \frac{1}{2}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\left(\color{blue}{y \cdot \left(4 \cdot \frac{1}{z}\right)} + 4 \cdot \frac{1}{2}\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(4 \cdot \frac{1}{z}\right) + \color{blue}{2}\right)\right) \]

   10. Applied rewrites 87.6%

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

Alternative 10: 34.0% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 -2.0)

C

double code(double x, double y, double z) {
	return -2.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return -2.0;
}

Python

def code(x, y, z):
	return -2.0

Julia

function code(x, y, z)
	return -2.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = -2.0;
end

Wolfram

code[x_, y_, z_] := -2.0

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{-2} \]
4. Step-by-step derivation

5. Applied rewrites 33.0%

   \[\leadsto \color{blue}{-2} \]
6. Add Preprocessing

Developer Target 1: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{x}{z} - \left(2 + 4 \cdot \frac{y}{z}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z)))))

C

double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * (x / z)) - (2.0d0 + (4.0d0 * (y / z)))
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Python

def code(x, y, z):
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)))

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(x / z)) - Float64(2.0 + Float64(4.0 * Float64(y / z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(2.0 + N[(4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024254 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* 4 (/ x z)) (+ 2 (* 4 (/ y z)))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))