Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 92.9% → 97.7%

Time: 7.5s

Alternatives: 9

Speedup: 1.1×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x y z t) :precision binary64 (fma (/ z t) (- y x) x))

C

double code(double x, double y, double z, double t) {
	return fma((z / t), (y - x), x);
}

Julia

function code(x, y, z, t)
	return fma(Float64(z / t), Float64(y - x), x)
end

Wolfram

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

Derivation

1. Initial program 92.5%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 97.5

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

4. Applied rewrites 97.5%

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

Alternative 2: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{z}{t}\\ t_2 := \frac{\left(y - x\right) \cdot z}{t} + x\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{y \cdot z}{t} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y x) (/ z t))) (t_2 (+ (/ (* (- y x) z) t) x)))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 2e+307) (+ (/ (* y z) t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - x) * (z / t);
	double t_2 = (((y - x) * z) / t) + x;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+307) {
		tmp = ((y * z) / t) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y - x) * (z / t);
	double t_2 = (((y - x) * z) / t) + x;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 2e+307) {
		tmp = ((y * z) / t) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y - x) * (z / t)
	t_2 = (((y - x) * z) / t) + x
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 2e+307:
		tmp = ((y * z) / t) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - x) * Float64(z / t))
	t_2 = Float64(Float64(Float64(Float64(y - x) * z) / t) + x)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+307)
		tmp = Float64(Float64(Float64(y * z) / t) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y - x) * (z / t);
	t_2 = (((y - x) * z) / t) + x;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 2e+307)
		tmp = ((y * z) / t) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 2e+307], N[(N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -inf.0 or 1.99999999999999997e307 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

   1. Initial program 79.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 79.7

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

   5. Applied rewrites 79.7%

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

   7. Applied rewrites 93.9%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 1.99999999999999997e307

   1. Initial program 98.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - x\right) \cdot z}{t} + x \leq -\infty:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;\frac{\left(y - x\right) \cdot z}{t} + x \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{y \cdot z}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(y - x\right) \cdot z}{t} + x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (/ (* (- y x) z) t) x) -5e-288) (* (/ y t) z) (/ (* y z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((((y - x) * z) / t) + x) <= -5e-288) {
		tmp = (y / t) * z;
	} else {
		tmp = (y * z) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((((y - x) * z) / t) + x) <= (-5d-288)) then
        tmp = (y / t) * z
    else
        tmp = (y * z) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((((y - x) * z) / t) + x) <= -5e-288) {
		tmp = (y / t) * z;
	} else {
		tmp = (y * z) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((((y - x) * z) / t) + x) <= -5e-288:
		tmp = (y / t) * z
	else:
		tmp = (y * z) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y - x) * z) / t) + x) <= -5e-288)
		tmp = Float64(Float64(y / t) * z);
	else
		tmp = Float64(Float64(y * z) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((((y - x) * z) / t) + x) <= -5e-288)
		tmp = (y / t) * z;
	else
		tmp = (y * z) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], -5e-288], N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision], N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -5.00000000000000011e-288

   1. Initial program 92.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 40.2

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

   5. Applied rewrites 40.2%

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

   if -5.00000000000000011e-288 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

   1. Initial program 92.8%

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

   3. Taylor expanded in y around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 37.7

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

   5. Applied rewrites 37.7%

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

   7. Applied rewrites 42.5%

      \[\leadsto \frac{y \cdot z}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - x\right) \cdot z}{t} + x \leq -5 \cdot 10^{-288}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{x}{t} \cdot z\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (* (/ x t) z))))
   (if (<= x -1.35e-71) t_1 (if (<= x 4e-11) (/ (* (- y x) z) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((x / t) * z);
	double tmp;
	if (x <= -1.35e-71) {
		tmp = t_1;
	} else if (x <= 4e-11) {
		tmp = ((y - x) * z) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x - ((x / t) * z)
    if (x <= (-1.35d-71)) then
        tmp = t_1
    else if (x <= 4d-11) then
        tmp = ((y - x) * z) / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - ((x / t) * z);
	double tmp;
	if (x <= -1.35e-71) {
		tmp = t_1;
	} else if (x <= 4e-11) {
		tmp = ((y - x) * z) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - ((x / t) * z)
	tmp = 0
	if x <= -1.35e-71:
		tmp = t_1
	elif x <= 4e-11:
		tmp = ((y - x) * z) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(x / t) * z))
	tmp = 0.0
	if (x <= -1.35e-71)
		tmp = t_1;
	elseif (x <= 4e-11)
		tmp = Float64(Float64(Float64(y - x) * z) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - ((x / t) * z);
	tmp = 0.0;
	if (x <= -1.35e-71)
		tmp = t_1;
	elseif (x <= 4e-11)
		tmp = ((y - x) * z) / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[(x / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e-71], t$95$1, If[LessEqual[x, 4e-11], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3500000000000001e-71 or 3.99999999999999976e-11 < x

   1. Initial program 91.5%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.9

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

   5. Applied rewrites 82.9%

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

   if -1.3500000000000001e-71 < x < 3.99999999999999976e-11

   1. Initial program 94.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 77.0

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

   5. Applied rewrites 77.0%

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

Alternative 5: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{x}{t} \cdot z\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-11}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (* (/ x t) z))))
   (if (<= x -1.35e-71) t_1 (if (<= x 3.5e-11) (* (- y x) (/ z t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((x / t) * z);
	double tmp;
	if (x <= -1.35e-71) {
		tmp = t_1;
	} else if (x <= 3.5e-11) {
		tmp = (y - x) * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x - ((x / t) * z)
    if (x <= (-1.35d-71)) then
        tmp = t_1
    else if (x <= 3.5d-11) then
        tmp = (y - x) * (z / t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - ((x / t) * z);
	double tmp;
	if (x <= -1.35e-71) {
		tmp = t_1;
	} else if (x <= 3.5e-11) {
		tmp = (y - x) * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - ((x / t) * z)
	tmp = 0
	if x <= -1.35e-71:
		tmp = t_1
	elif x <= 3.5e-11:
		tmp = (y - x) * (z / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(x / t) * z))
	tmp = 0.0
	if (x <= -1.35e-71)
		tmp = t_1;
	elseif (x <= 3.5e-11)
		tmp = Float64(Float64(y - x) * Float64(z / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - ((x / t) * z);
	tmp = 0.0;
	if (x <= -1.35e-71)
		tmp = t_1;
	elseif (x <= 3.5e-11)
		tmp = (y - x) * (z / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[(x / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e-71], t$95$1, If[LessEqual[x, 3.5e-11], N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3500000000000001e-71 or 3.50000000000000019e-11 < x

   1. Initial program 91.5%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.9

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

   5. Applied rewrites 82.9%

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

   if -1.3500000000000001e-71 < x < 3.50000000000000019e-11

   1. Initial program 94.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 77.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-71}:\\ \;\;\;\;x - \frac{x}{t} \cdot z\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-11}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-x}{t} \cdot z\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (- x) t) z)))
   (if (<= x -8.8e+18) t_1 (if (<= x 2.3e+68) (* y (/ z t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (-x / t) * z;
	double tmp;
	if (x <= -8.8e+18) {
		tmp = t_1;
	} else if (x <= 2.3e+68) {
		tmp = y * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-x / t) * z
    if (x <= (-8.8d+18)) then
        tmp = t_1
    else if (x <= 2.3d+68) then
        tmp = y * (z / t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (-x / t) * z;
	double tmp;
	if (x <= -8.8e+18) {
		tmp = t_1;
	} else if (x <= 2.3e+68) {
		tmp = y * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (-x / t) * z
	tmp = 0
	if x <= -8.8e+18:
		tmp = t_1
	elif x <= 2.3e+68:
		tmp = y * (z / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(-x) / t) * z)
	tmp = 0.0
	if (x <= -8.8e+18)
		tmp = t_1;
	elseif (x <= 2.3e+68)
		tmp = Float64(y * Float64(z / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (-x / t) * z;
	tmp = 0.0;
	if (x <= -8.8e+18)
		tmp = t_1;
	elseif (x <= 2.3e+68)
		tmp = y * (z / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[((-x) / t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[x, -8.8e+18], t$95$1, If[LessEqual[x, 2.3e+68], N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.8e18 or 2.3e68 < x

   1. Initial program 89.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 47.3

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 42.5%

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

   if -8.8e18 < x < 2.3e68

   1. Initial program 94.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 62.2

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

   7. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\frac{z}{t} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{-x}{t} \cdot z\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.5%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 57.7

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

5. Applied rewrites 57.7%

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

7. Applied rewrites 60.8%

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

8. Final simplification 60.8%

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

Alternative 8: 40.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ y \cdot \frac{z}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* y (/ z t)))

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y * (z / t)
end function

Java

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

Python

def code(x, y, z, t):
	return y * (z / t)

Julia

function code(x, y, z, t)
	return Float64(y * Float64(z / t))
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.5%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 97.5

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

4. Applied rewrites 97.5%

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

5. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 42.1

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

7. Applied rewrites 42.1%

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

8. Final simplification 42.1%

   \[\leadsto y \cdot \frac{z}{t} \]
9. Add Preprocessing

Alternative 9: 37.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{y}{t} \cdot z \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (/ y t) z))

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y / t) * z
end function

Java

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

Python

def code(x, y, z, t):
	return (y / t) * z

Julia

function code(x, y, z, t)
	return Float64(Float64(y / t) * z)
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 92.5%

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

3. Taylor expanded in y around inf

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 39.0

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

5. Applied rewrites 39.0%

   \[\leadsto \color{blue}{\frac{y}{t} \cdot z} \]
6. Add Preprocessing

Developer Target 1: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< x -9.025511195533005e-135)
   (- x (* (/ z t) (- x y)))
   (if (< x 4.275032163700715e-250)
     (+ x (* (/ (- y x) t) z))
     (+ x (/ (- y x) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x < (-9.025511195533005d-135)) then
        tmp = x - ((z / t) * (x - y))
    else if (x < 4.275032163700715d-250) then
        tmp = x + (((y - x) / t) * z)
    else
        tmp = x + ((y - x) / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x < -9.025511195533005e-135:
		tmp = x - ((z / t) * (x - y))
	elif x < 4.275032163700715e-250:
		tmp = x + (((y - x) / t) * z)
	else:
		tmp = x + ((y - x) / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x < -9.025511195533005e-135)
		tmp = Float64(x - Float64(Float64(z / t) * Float64(x - y)));
	elseif (x < 4.275032163700715e-250)
		tmp = Float64(x + Float64(Float64(Float64(y - x) / t) * z));
	else
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x < -9.025511195533005e-135)
		tmp = x - ((z / t) * (x - y));
	elseif (x < 4.275032163700715e-250)
		tmp = x + (((y - x) / t) * z);
	else
		tmp = x + ((y - x) / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[x, -9.025511195533005e-135], N[(x - N[(N[(z / t), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[x, 4.275032163700715e-250], N[(x + N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024270 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -1805102239106601/200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- x (* (/ z t) (- x y))) (if (< x 855006432740143/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z))))))

  (+ x (/ (* (- y x) z) t)))